modular techniques in computational algebraic geometry · modular techniques in computational...
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![Page 1: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/1.jpg)
Modular Techniques in Computational AlgebraicGeometry
Janko Boehmjoint with W. Decker, C. Fieker, S. Laplagne, G. Pfister
Technische Universitat Kaiserslautern
01 October 2015
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 2: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/2.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 3: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/3.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 4: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/4.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.
2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 5: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/5.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 6: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/6.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.
Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 7: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/7.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 8: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/8.jpg)
Modular computations
Many exact computations in computer algebra are carried out over Q
and extensions thereof.
Modular techniques are an important tool to improve performance ofalgorithms over Q.
Fundamental approach:
1 Compute modulo primes.2 Reconstruct result over Q.
Benefits:
Avoid intermediate coefficient growth.Obtain parallel version of the algorithm.
Goal:General reconstruction scheme for algorithms in commutative algebra,algebraic geometry, number theory.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 0 / 26
![Page 9: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/9.jpg)
Outline
Modular computations and rational reconstruction
Bad primes
Error tolerant lifting
General reconstruction scheme
Normalization
Local-to-global algorithm for adjoint ideals
Modular version and verification
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 1 / 26
![Page 10: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/10.jpg)
Outline
Modular computations and rational reconstruction
Bad primes
Error tolerant lifting
General reconstruction scheme
Normalization
Local-to-global algorithm for adjoint ideals
Modular version and verification
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 1 / 26
![Page 11: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/11.jpg)
Outline
Modular computations and rational reconstruction
Bad primes
Error tolerant lifting
General reconstruction scheme
Normalization
Local-to-global algorithm for adjoint ideals
Modular version and verification
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 1 / 26
![Page 12: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/12.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 ) 7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 13: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/13.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 ) 7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 14: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/14.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 ) 7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 15: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/15.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 )
7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 16: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/16.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 ) 7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 17: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/17.jpg)
Modular computations
Example
Compute3
4+
1
3=
13
12
using modular techniques:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
34 7→ ( 2 , 6 , 9 , 26 )
+
13 7→ ( 2 , 5 , 4 , 34 )
q
( 4 , 4 , 2 , 60 ) 7→ 22684
How to obtain a rational number from 22684?
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 2 / 26
![Page 18: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/18.jpg)
Rational reconstruction
Theorem (Kornerup, Gregory, 1983)
The Farey map{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b,N) = 1
|a| , |b| ≤√(N − 1)/2
}−→ Z/N
ab 7−→ a · b−1
is injective. Efficient algorithm for preimage.
Example
Indeed, in the above example{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b, 38885) = 1
|a| , |b| ≤ 139
}−→ Z/38885
1312 7−→ 22684
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 3 / 26
![Page 19: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/19.jpg)
Rational reconstruction
Theorem (Kornerup, Gregory, 1983)
The Farey map{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b,N) = 1
|a| , |b| ≤√(N − 1)/2
}−→ Z/N
ab 7−→ a · b−1
is injective.
Efficient algorithm for preimage.
Example
Indeed, in the above example{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b, 38885) = 1
|a| , |b| ≤ 139
}−→ Z/38885
1312 7−→ 22684
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 3 / 26
![Page 20: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/20.jpg)
Rational reconstruction
Theorem (Kornerup, Gregory, 1983)
The Farey map{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b,N) = 1
|a| , |b| ≤√(N − 1)/2
}−→ Z/N
ab 7−→ a · b−1
is injective. Efficient algorithm for preimage.
Example
Indeed, in the above example{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b, 38885) = 1
|a| , |b| ≤ 139
}−→ Z/38885
1312 7−→ 22684
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 3 / 26
![Page 21: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/21.jpg)
Rational reconstruction
Theorem (Kornerup, Gregory, 1983)
The Farey map{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b,N) = 1
|a| , |b| ≤√(N − 1)/2
}−→ Z/N
ab 7−→ a · b−1
is injective. Efficient algorithm for preimage.
Example
Indeed, in the above example{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b, 38885) = 1
|a| , |b| ≤ 139
}−→ Z/38885
1312 7−→ 22684
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 3 / 26
![Page 22: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/22.jpg)
Rational reconstruction
Theorem (Kornerup, Gregory, 1983)
The Farey map{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b,N) = 1
|a| , |b| ≤√(N − 1)/2
}−→ Z/N
ab 7−→ a · b−1
is injective. Efficient algorithm for preimage.
Example
Indeed, in the above example{ab ∈ Q
∣∣∣∣ gcd(a, b) = 1gcd(b, 38885) = 1
|a| , |b| ≤ 139
}−→ Z/38885
1312 7−→ 22684
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 3 / 26
![Page 23: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/23.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 24: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/24.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 25: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/25.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 26: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/26.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 27: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/27.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 28: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/28.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 29: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/29.jpg)
Basic concept for modular computations
1 Compute result over Z/pi for distinct primes p1, . . . , pr .
2 For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism
Z/N ∼= Z/p1 × . . .×Z/pr
3 If exists, compute preimage w.r.t injective Farey map.
4 Verify correctness of lift.
This will yield correct result, provided
N is large enough s.t. the Q-result is in source of Farey map, and
none of the pi is bad.
Definition
A prime p is called bad if the result over Q does not reduce modulo p tothe result over Z/p.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 4 / 26
![Page 30: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/30.jpg)
Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G .
For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
![Page 31: Modular Techniques in Computational Algebraic Geometry · Modular Techniques in Computational Algebraic Geometry Janko Boehm joint with W. Decker, C. Fieker, S. Laplagne, G. P ster](https://reader033.vdocuments.net/reader033/viewer/2022042521/5f6a99aab745bb01161261cc/html5/thumbnails/31.jpg)
Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve,
and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ].
Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
For G ⊂ K [X ] = K [x1, . . . , xn] and a monomial ordering >, let LM(G ) bethe set of lead monomials of G . For G ⊂ Z[X ] define
Gp := G ⊂ Z/p [X ].
Theorem (Arnold, 2003)
Suppose F = {f1, ..., fr} ⊂ Z[X ] with fi primitve, and
G is the reduced Grobner basis of 〈F 〉 ⊂ Q[X ],
G (p) is the reduced Grobner basis of 〈Fp〉, and
GZ a minimal strong Grobnerbasis of 〈F 〉 ⊂ Z[X ]. Then
p does not divide any lead coefficient in GZ ⇐⇒ LMG = LMG (p)
⇐⇒ Gp = G (p)
that is, p is not bad.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 5 / 26
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Bad primes in Grobner basis computations
Example
Letf = x5 + y11 + xy9 + x3y9 ∈ Z[x , y ].
Then GZ for ⟨∂f
∂x,
∂f
∂y
⟩w.r.t lp is
264627y39 + . . . ,
12103947791971846719838321886393392913750065060875xy8 − . . . ,
40754032969602177507873137664624218564815033875x4 + . . . .
and LMG = LMG (p) for all primes p except
p = 3, 5, 11, 809, 65179, 531264751, 431051934846786628615463393.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 6 / 26
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Bad primes in Grobner basis computations
Example
Letf = x5 + y11 + xy9 + x3y9 ∈ Z[x , y ].
Then GZ for ⟨∂f
∂x,
∂f
∂y
⟩w.r.t lp is
264627y39 + . . . ,
12103947791971846719838321886393392913750065060875xy8 − . . . ,
40754032969602177507873137664624218564815033875x4 + . . . .
and LMG = LMG (p) for all primes p except
p = 3, 5, 11, 809, 65179, 531264751, 431051934846786628615463393.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 6 / 26
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Bad primes in Grobner basis computations
Example
Letf = x5 + y11 + xy9 + x3y9 ∈ Z[x , y ].
Then GZ for ⟨∂f
∂x,
∂f
∂y
⟩w.r.t lp is
264627y39 + . . . ,
12103947791971846719838321886393392913750065060875xy8 − . . . ,
40754032969602177507873137664624218564815033875x4 + . . . .
and LMG = LMG (p) for all primes p except
p = 3, 5, 11, 809, 65179, 531264751, 431051934846786628615463393.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 6 / 26
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Bad primes
Classification of bad primes:
Type 1: Input modulo p not valid (no problem)
Type 2: Failure in the course of the algorithm (e.g. matrix notinvertible modulo p, wastes computation time if happens)
Type 3: Computable invariant with known expected value (e.g.Hilbert polynomial) is wrong (have to do expensive test for eachprime, although set of bad primes usually is finite)
Type 4: Computable invariant with unknown expected value (e.g.lead ideal in Grobner basis computations) is wrong (to detect by amajority vote, have to compute and store value of invariant for allmodular results)
Type 5: otherwise.
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Bad primes
Classification of bad primes:
Type 1: Input modulo p not valid (no problem)
Type 2: Failure in the course of the algorithm (e.g. matrix notinvertible modulo p, wastes computation time if happens)
Type 3: Computable invariant with known expected value (e.g.Hilbert polynomial) is wrong (have to do expensive test for eachprime, although set of bad primes usually is finite)
Type 4: Computable invariant with unknown expected value (e.g.lead ideal in Grobner basis computations) is wrong (to detect by amajority vote, have to compute and store value of invariant for allmodular results)
Type 5: otherwise.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 7 / 26
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Bad primes
Classification of bad primes:
Type 1: Input modulo p not valid (no problem)
Type 2: Failure in the course of the algorithm (e.g. matrix notinvertible modulo p, wastes computation time if happens)
Type 3: Computable invariant with known expected value (e.g.Hilbert polynomial) is wrong (have to do expensive test for eachprime, although set of bad primes usually is finite)
Type 4: Computable invariant with unknown expected value (e.g.lead ideal in Grobner basis computations) is wrong (to detect by amajority vote, have to compute and store value of invariant for allmodular results)
Type 5: otherwise.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 7 / 26
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Bad primes
Classification of bad primes:
Type 1: Input modulo p not valid (no problem)
Type 2: Failure in the course of the algorithm (e.g. matrix notinvertible modulo p, wastes computation time if happens)
Type 3: Computable invariant with known expected value (e.g.Hilbert polynomial) is wrong (have to do expensive test for eachprime, although set of bad primes usually is finite)
Type 4: Computable invariant with unknown expected value (e.g.lead ideal in Grobner basis computations) is wrong (to detect by amajority vote, have to compute and store value of invariant for allmodular results)
Type 5: otherwise.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 7 / 26
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Bad primes
Classification of bad primes:
Type 1: Input modulo p not valid (no problem)
Type 2: Failure in the course of the algorithm (e.g. matrix notinvertible modulo p, wastes computation time if happens)
Type 3: Computable invariant with known expected value (e.g.Hilbert polynomial) is wrong (have to do expensive test for eachprime, although set of bad primes usually is finite)
Type 4: Computable invariant with unknown expected value (e.g.lead ideal in Grobner basis computations) is wrong (to detect by amajority vote, have to compute and store value of invariant for allmodular results)
Type 5: otherwise.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 7 / 26
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Example of type 5 bad prime
For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩Z[X ])p.
Example
Consider the algorithm I 7→√
I + Jac(I ) for
I =⟨x6 + y6 + 7x5z + x3y2z − 31x4z2 − 224x3z3 + 244x2z4 + 1632xz5 + 576z6
⟩
Then w.r.t dp LM(I ) =⟨x6⟩= LM(I5)
U(0) =√
I + Jac(I ) = 〈y , x − 4z〉 ∩ 〈y , x + 6z〉
U(5) =√
I5 + Jac(I5) =⟨y , x2 − z2
⟩= 〈y , x − z〉 ∩ 〈y , x + z〉
U(0)5 =⟨y , (x + z)2
⟩Hence
U(0)5 6= U(5)
LM(U(0)) =⟨y , x2
⟩= LM(U(5))
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 8 / 26
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Example of type 5 bad prime
For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩Z[X ])p.
Example
Consider the algorithm I 7→√
I + Jac(I ) for
I =⟨x6 + y6 + 7x5z + x3y2z − 31x4z2 − 224x3z3 + 244x2z4 + 1632xz5 + 576z6
⟩Then w.r.t dp LM(I ) =
⟨x6⟩= LM(I5)
U(0) =√
I + Jac(I ) = 〈y , x − 4z〉 ∩ 〈y , x + 6z〉
U(5) =√
I5 + Jac(I5) =⟨y , x2 − z2
⟩= 〈y , x − z〉 ∩ 〈y , x + z〉
U(0)5 =⟨y , (x + z)2
⟩Hence
U(0)5 6= U(5)
LM(U(0)) =⟨y , x2
⟩= LM(U(5))
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 8 / 26
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Example of type 5 bad prime
For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩Z[X ])p.
Example
Consider the algorithm I 7→√
I + Jac(I ) for
I =⟨x6 + y6 + 7x5z + x3y2z − 31x4z2 − 224x3z3 + 244x2z4 + 1632xz5 + 576z6
⟩Then w.r.t dp LM(I ) =
⟨x6⟩= LM(I5)
U(0) =√
I + Jac(I ) = 〈y , x − 4z〉 ∩ 〈y , x + 6z〉
U(5) =√
I5 + Jac(I5) =⟨y , x2 − z2
⟩= 〈y , x − z〉 ∩ 〈y , x + z〉
U(0)5 =⟨y , (x + z)2
⟩Hence
U(0)5 6= U(5)
LM(U(0)) =⟨y , x2
⟩= LM(U(5))
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 8 / 26
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Example of type 5 bad prime
For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩Z[X ])p.
Example
Consider the algorithm I 7→√
I + Jac(I ) for
I =⟨x6 + y6 + 7x5z + x3y2z − 31x4z2 − 224x3z3 + 244x2z4 + 1632xz5 + 576z6
⟩Then w.r.t dp LM(I ) =
⟨x6⟩= LM(I5)
U(0) =√
I + Jac(I ) = 〈y , x − 4z〉 ∩ 〈y , x + 6z〉
U(5) =√
I5 + Jac(I5) =⟨y , x2 − z2
⟩= 〈y , x − z〉 ∩ 〈y , x + z〉
U(0)5 =⟨y , (x + z)2
⟩
HenceU(0)5 6= U(5)
LM(U(0)) =⟨y , x2
⟩= LM(U(5))
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 8 / 26
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Example of type 5 bad prime
For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩Z[X ])p.
Example
Consider the algorithm I 7→√
I + Jac(I ) for
I =⟨x6 + y6 + 7x5z + x3y2z − 31x4z2 − 224x3z3 + 244x2z4 + 1632xz5 + 576z6
⟩Then w.r.t dp LM(I ) =
⟨x6⟩= LM(I5)
U(0) =√
I + Jac(I ) = 〈y , x − 4z〉 ∩ 〈y , x + 6z〉
U(5) =√
I5 + Jac(I5) =⟨y , x2 − z2
⟩= 〈y , x − z〉 ∩ 〈y , x + z〉
U(0)5 =⟨y , (x + z)2
⟩Hence
U(0)5 6= U(5)
LM(U(0)) =⟨y , x2
⟩= LM(U(5))
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 8 / 26
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Error tolerant reconstruction
Goal: Reconstruct ab from r ∈ Z/N in the presence of bad primes.
Idea: Find (x , y) with xy = a
b in the lattice
Λ = 〈(N, 0), (r , 1)〉 ⊂ Z2
Lemma (BDFP, 2015)
All (x , y) ∈ Λ with x2 + y2 < N are collinear.
Proof.
Let λ = (x , y), µ = (c, d) ∈ Λ with x2 + y2, c2 + d2 < N. Thenyµ− dλ = (yc − xd , 0) ∈ Λ, so N |(yc − xd). By Cauchy–Schwarz|yc − xd | < N, hence yc = xd .
Now supposeN = N ′ ·M
with gcd(N ′,M) = 1.
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Error tolerant reconstruction
Goal: Reconstruct ab from r ∈ Z/N in the presence of bad primes.
Idea: Find (x , y) with xy = a
b in the lattice
Λ = 〈(N, 0), (r , 1)〉 ⊂ Z2
Lemma (BDFP, 2015)
All (x , y) ∈ Λ with x2 + y2 < N are collinear.
Proof.
Let λ = (x , y), µ = (c, d) ∈ Λ with x2 + y2, c2 + d2 < N. Thenyµ− dλ = (yc − xd , 0) ∈ Λ, so N |(yc − xd). By Cauchy–Schwarz|yc − xd | < N, hence yc = xd .
Now supposeN = N ′ ·M
with gcd(N ′,M) = 1.
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Error tolerant reconstruction
Goal: Reconstruct ab from r ∈ Z/N in the presence of bad primes.
Idea: Find (x , y) with xy = a
b in the lattice
Λ = 〈(N, 0), (r , 1)〉 ⊂ Z2
Lemma (BDFP, 2015)
All (x , y) ∈ Λ with x2 + y2 < N are collinear.
Proof.
Let λ = (x , y), µ = (c, d) ∈ Λ with x2 + y2, c2 + d2 < N. Thenyµ− dλ = (yc − xd , 0) ∈ Λ, so N |(yc − xd). By Cauchy–Schwarz|yc − xd | < N, hence yc = xd .
Now supposeN = N ′ ·M
with gcd(N ′,M) = 1.
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Error tolerant reconstruction
Goal: Reconstruct ab from r ∈ Z/N in the presence of bad primes.
Idea: Find (x , y) with xy = a
b in the lattice
Λ = 〈(N, 0), (r , 1)〉 ⊂ Z2
Lemma (BDFP, 2015)
All (x , y) ∈ Λ with x2 + y2 < N are collinear.
Proof.
Let λ = (x , y), µ = (c, d) ∈ Λ with x2 + y2, c2 + d2 < N. Thenyµ− dλ = (yc − xd , 0) ∈ Λ, so N |(yc − xd). By Cauchy–Schwarz|yc − xd | < N, hence yc = xd .
Now supposeN = N ′ ·M
with gcd(N ′,M) = 1.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 9 / 26
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Error tolerant reconstruction
Goal: Reconstruct ab from r ∈ Z/N in the presence of bad primes.
Idea: Find (x , y) with xy = a
b in the lattice
Λ = 〈(N, 0), (r , 1)〉 ⊂ Z2
Lemma (BDFP, 2015)
All (x , y) ∈ Λ with x2 + y2 < N are collinear.
Proof.
Let λ = (x , y), µ = (c, d) ∈ Λ with x2 + y2, c2 + d2 < N. Thenyµ− dλ = (yc − xd , 0) ∈ Λ, so N |(yc − xd). By Cauchy–Schwarz|yc − xd | < N, hence yc = xd .
Now supposeN = N ′ ·M
with gcd(N ′,M) = 1.Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 9 / 26
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Error tolerant reconstruction
Think of N ′ as the product of the good primes with correct result s,and of M as the product of the bad primes with wrong result t.
Theorem (BDFP, 2015)
Ifr 7→ (s, t) with respect to Z/N ∼= Z/N ′ ×Z/M
and a
b≡ s modN ′
then (aM, bM) ∈ Λ. So if
(a2 + b2)M < N ′,
then (by the lemma)
x
y=
a
bfor all (x , y) ∈ Λ with (x2 + y2) < N
and such vectors exist. Moreover, if gcd(a, b) = 1 and (x , y) is a shortestvector 6= 0 in Λ, we also have gcd(x , y)|M.
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Error tolerant reconstruction
Think of N ′ as the product of the good primes with correct result s,and of M as the product of the bad primes with wrong result t.
Theorem (BDFP, 2015)
Ifr 7→ (s, t) with respect to Z/N ∼= Z/N ′ ×Z/M
and a
b≡ s modN ′
then (aM, bM) ∈ Λ. So if
(a2 + b2)M < N ′,
then (by the lemma)
x
y=
a
bfor all (x , y) ∈ Λ with (x2 + y2) < N
and such vectors exist. Moreover, if gcd(a, b) = 1 and (x , y) is a shortestvector 6= 0 in Λ, we also have gcd(x , y)|M.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 10 / 26
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Error tolerant reconstruction
Think of N ′ as the product of the good primes with correct result s,and of M as the product of the bad primes with wrong result t.
Theorem (BDFP, 2015)
Ifr 7→ (s, t) with respect to Z/N ∼= Z/N ′ ×Z/M
and a
b≡ s modN ′
then (aM, bM) ∈ Λ.
So if
(a2 + b2)M < N ′,
then (by the lemma)
x
y=
a
bfor all (x , y) ∈ Λ with (x2 + y2) < N
and such vectors exist. Moreover, if gcd(a, b) = 1 and (x , y) is a shortestvector 6= 0 in Λ, we also have gcd(x , y)|M.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 10 / 26
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Error tolerant reconstruction
Think of N ′ as the product of the good primes with correct result s,and of M as the product of the bad primes with wrong result t.
Theorem (BDFP, 2015)
Ifr 7→ (s, t) with respect to Z/N ∼= Z/N ′ ×Z/M
and a
b≡ s modN ′
then (aM, bM) ∈ Λ. So if
(a2 + b2)M < N ′,
then (by the lemma)
x
y=
a
bfor all (x , y) ∈ Λ with (x2 + y2) < N
and such vectors exist.
Moreover, if gcd(a, b) = 1 and (x , y) is a shortestvector 6= 0 in Λ, we also have gcd(x , y)|M.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 10 / 26
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Error tolerant reconstruction
Think of N ′ as the product of the good primes with correct result s,and of M as the product of the bad primes with wrong result t.
Theorem (BDFP, 2015)
Ifr 7→ (s, t) with respect to Z/N ∼= Z/N ′ ×Z/M
and a
b≡ s modN ′
then (aM, bM) ∈ Λ. So if
(a2 + b2)M < N ′,
then (by the lemma)
x
y=
a
bfor all (x , y) ∈ Λ with (x2 + y2) < N
and such vectors exist. Moreover, if gcd(a, b) = 1 and (x , y) is a shortestvector 6= 0 in Λ, we also have gcd(x , y)|M.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 10 / 26
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Error tolerant reconstruction via Gauss-Lagrange
Hence, if N ′ � M, the Gauss-Lagrange-Algorithm for finding a shortestvector (x , y) ∈ Λ gives a
b independently of t, provided x2 + y2 < N.
Algorithm (Error tolerant reconstruction)
Input: N and r .Output: a
b or false.1: (a0, b0) := (N, 0), (a1, b1) := (r , 1), i := −12: repeat3: i = i + 1
4: (ai+2, bi+2) = (ai , bi )−⌊〈(ai , bi ), (ai+1, bi+1)〉‖(ai+1, bi+1)‖2
⌉(ai+1, bi+1)
5: until a2i+2 + b2i+2 ≥ a2i+1 + b2i+1
6: if a2i+1 + b2i+1 < N then7: return ai+1
bi+1
8: else9: return false
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Error tolerant reconstruction via Gauss-Lagrange
Hence, if N ′ � M, the Gauss-Lagrange-Algorithm for finding a shortestvector (x , y) ∈ Λ gives a
b independently of t, provided x2 + y2 < N.
Algorithm (Error tolerant reconstruction)
Input: N and r .Output: a
b or false.1: (a0, b0) := (N, 0), (a1, b1) := (r , 1), i := −12: repeat3: i = i + 1
4: (ai+2, bi+2) = (ai , bi )−⌊〈(ai , bi ), (ai+1, bi+1)〉‖(ai+1, bi+1)‖2
⌉(ai+1, bi+1)
5: until a2i+2 + b2i+2 ≥ a2i+1 + b2i+1
6: if a2i+1 + b2i+1 < N then7: return ai+1
bi+1
8: else9: return false
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Reconstruction via Gauss-Lagrange
Example
We reconstruct 1312 from
22684 ∈ Z/38885
by determining a shortest vector in the lattice
〈(38885, 0), (22684, 1)〉 ⊂ Z2
via Gauss-Lagrange
(38885, 0) = 2 · (22684, 1) + (−6483,−2),
(22684, 1) = −3 · (−6483,−2) + (3235,−5),
(−6483,−2) = 2 · (3235,−5) + (−13,−12),
(3235,−5) = −134 · (−13,−12) + (1493,−1613).
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 12 / 26
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Reconstruction via Gauss-Lagrange
Example
We reconstruct 1312 from
22684 ∈ Z/38885
by determining a shortest vector in the lattice
〈(38885, 0), (22684, 1)〉 ⊂ Z2
via Gauss-Lagrange
(38885, 0) = 2 · (22684, 1) + (−6483,−2),
(22684, 1) = −3 · (−6483,−2) + (3235,−5),
(−6483,−2) = 2 · (3235,−5) + (−13,−12),
(3235,−5) = −134 · (−13,−12) + (1493,−1613).
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 12 / 26
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Reconstruction via Gauss-Lagrange
Example
Now introduce an error in the modular results:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
( 4 , 4 , 2 , 60 ) 7→ 22684
( 4 , 2 , 2 60 ) 7→ 464
Error tolerant reconstruction computes
(38885, 0) = 84 · (464, 1) + (−91,−84),
(464, 1) = −3 · (−91,−84) + (191,−251)
hence yields91
84=
7 · 13
7 · 12=
13
12.
Note that(132 + 122) · 7 = 2191 < 5555 = 5 · 11 · 101.
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Reconstruction via Gauss-Lagrange
Example
Now introduce an error in the modular results:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
( 4 , 4 , 2 , 60 ) 7→ 22684
( 4 , 2 , 2 60 ) 7→ 464
Error tolerant reconstruction computes
(38885, 0) = 84 · (464, 1) + (−91,−84),
(464, 1) = −3 · (−91,−84) + (191,−251)
hence yields91
84=
7 · 13
7 · 12=
13
12.
Note that(132 + 122) · 7 = 2191 < 5555 = 5 · 11 · 101.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 13 / 26
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Reconstruction via Gauss-Lagrange
Example
Now introduce an error in the modular results:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
( 4 , 4 , 2 , 60 ) 7→ 22684
( 4 , 2 , 2 60 ) 7→ 464
Error tolerant reconstruction computes
(38885, 0) = 84 · (464, 1) + (−91,−84),
(464, 1) = −3 · (−91,−84) + (191,−251)
hence yields91
84=
7 · 13
7 · 12=
13
12.
Note that(132 + 122) · 7 = 2191 < 5555 = 5 · 11 · 101.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 13 / 26
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Reconstruction via Gauss-Lagrange
Example
Now introduce an error in the modular results:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
( 4 , 4 , 2 , 60 ) 7→ 22684
( 4 , 2 , 2 60 ) 7→ 464
Error tolerant reconstruction computes
(38885, 0) = 84 · (464, 1) + (−91,−84),
(464, 1) = −3 · (−91,−84) + (191,−251)
hence yields91
84=
7 · 13
7 · 12=
13
12.
Note that(132 + 122) · 7 = 2191 < 5555 = 5 · 11 · 101.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 13 / 26
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Reconstruction via Gauss-Lagrange
Example
Now introduce an error in the modular results:
Z/5 × Z/7 × Z/11 × Z/101 ∼= Z/38885
( 4 , 4 , 2 , 60 ) 7→ 22684
( 4 , 2 , 2 60 ) 7→ 464
Error tolerant reconstruction computes
(38885, 0) = 84 · (464, 1) + (−91,−84),
(464, 1) = −3 · (−91,−84) + (191,−251)
hence yields91
84=
7 · 13
7 · 12=
13
12.
Note that(132 + 122) · 7 = 2191 < 5555 = 5 · 11 · 101.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 13 / 26
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 14 / 26
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 14 / 26
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 14 / 26
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 14 / 26
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General reconstruction scheme
Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U(0) associated toI by deterministic algorithm.
Algorithm
For Ip compute result U(p) over Z/p for p in finite set of primes P .
Reduce P according to majority vote on LM(U(p)).
For N = ∏p∈P p compute termwise CRT–lift U(N) to Z/N.
Lift U(N) by error tolerant rational reconstruction to U.
Test Up = U(p) for random prime p.
Verify U = U(0).
If lift, test or verification fails, then enlarge P .
Theorem (BDFP, 2015)
If the bad primes form a Zariski closed true subset of Spec Z, then thisalgorithm terminates with the correct result.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 14 / 26
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Normalization
Setup: A = K [X ]/I domain.
Definition
The normalization A of A is the integral closure of A in its quotient fieldQ(A).
We call A normal if A = A.
Theorem (Noether)
A is a finitely generated A-module.
Example
Curve I =⟨x3 + x2 − y2
⟩⊂ K [x , y ]
A = K [x , y ]/I ∼= K [t2 − 1, t3 − t] ⊂ K [t] ∼= Ax 7→ t2 − 1y 7→ t3 − t
As an A-module A =⟨
1, yx
⟩.
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Normalization
Setup: A = K [X ]/I domain.
Definition
The normalization A of A is the integral closure of A in its quotient fieldQ(A). We call A normal if A = A.
Theorem (Noether)
A is a finitely generated A-module.
Example
Curve I =⟨x3 + x2 − y2
⟩⊂ K [x , y ]
A = K [x , y ]/I ∼= K [t2 − 1, t3 − t] ⊂ K [t] ∼= Ax 7→ t2 − 1y 7→ t3 − t
As an A-module A =⟨
1, yx
⟩.
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Normalization
Setup: A = K [X ]/I domain.
Definition
The normalization A of A is the integral closure of A in its quotient fieldQ(A). We call A normal if A = A.
Theorem (Noether)
A is a finitely generated A-module.
Example
Curve I =⟨x3 + x2 − y2
⟩⊂ K [x , y ]
A = K [x , y ]/I ∼= K [t2 − 1, t3 − t] ⊂ K [t] ∼= Ax 7→ t2 − 1y 7→ t3 − t
As an A-module A =⟨
1, yx
⟩.
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Normalization
Setup: A = K [X ]/I domain.
Definition
The normalization A of A is the integral closure of A in its quotient fieldQ(A). We call A normal if A = A.
Theorem (Noether)
A is a finitely generated A-module.
Example
Curve I =⟨x3 + x2 − y2
⟩⊂ K [x , y ]
A = K [x , y ]/I ∼= K [t2 − 1, t3 − t] ⊂ K [t] ∼= Ax 7→ t2 − 1y 7→ t3 − t
As an A-module A =⟨
1, yx
⟩.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 15 / 26
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Normalization
Setup: A = K [X ]/I domain.
Definition
The normalization A of A is the integral closure of A in its quotient fieldQ(A). We call A normal if A = A.
Theorem (Noether)
A is a finitely generated A-module.
Example
Curve I =⟨x3 + x2 − y2
⟩⊂ K [x , y ]
A = K [x , y ]/I ∼= K [t2 − 1, t3 − t] ⊂ K [t] ∼= Ax 7→ t2 − 1y 7→ t3 − t
As an A-module A =⟨
1, yx
⟩.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 15 / 26
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Normalization
Lemma
If J ⊂ A is an ideal and 0 6= g ∈ J, then
A ↪→ HomA(J, J) ∼= 1g (gJ :A J) ⊂ A
a 7→ a·ϕ 7→ ϕ(g )
g
Algorithm
Starting from A0 = A and J0 = J, setting
Ai+1 =1g (gJi :Ai
Ji ) Ji =√JAi
we get a chain of extensions of reduced Noetherian rings
A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1.
Terminates since A is Noetherian.
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Normalization
Lemma
If J ⊂ A is an ideal and 0 6= g ∈ J, then
A ↪→ HomA(J, J) ∼= 1g (gJ :A J) ⊂ A
a 7→ a·ϕ 7→ ϕ(g )
g
Algorithm
Starting from A0 = A and J0 = J, setting
Ai+1 =1g (gJi :Ai
Ji ) Ji =√JAi
we get a chain of extensions of reduced Noetherian rings
A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1.
Terminates since A is Noetherian.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 16 / 26
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Normalization
Lemma
If J ⊂ A is an ideal and 0 6= g ∈ J, then
A ↪→ HomA(J, J) ∼= 1g (gJ :A J) ⊂ A
a 7→ a·ϕ 7→ ϕ(g )
g
Algorithm
Starting from A0 = A and J0 = J,
setting
Ai+1 =1g (gJi :Ai
Ji ) Ji =√JAi
we get a chain of extensions of reduced Noetherian rings
A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1.
Terminates since A is Noetherian.
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Normalization
Lemma
If J ⊂ A is an ideal and 0 6= g ∈ J, then
A ↪→ HomA(J, J) ∼= 1g (gJ :A J) ⊂ A
a 7→ a·ϕ 7→ ϕ(g )
g
Algorithm
Starting from A0 = A and J0 = J, setting
Ai+1 =1g (gJi :Ai
Ji ) Ji =√JAi
we get a chain of extensions of reduced Noetherian rings
A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1.
Terminates since A is Noetherian.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 16 / 26
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Normalization
Lemma
If J ⊂ A is an ideal and 0 6= g ∈ J, then
A ↪→ HomA(J, J) ∼= 1g (gJ :A J) ⊂ A
a 7→ a·ϕ 7→ ϕ(g )
g
Algorithm
Starting from A0 = A and J0 = J, setting
Ai+1 =1g (gJi :Ai
Ji ) Ji =√JAi
we get a chain of extensions of reduced Noetherian rings
A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1.
Terminates since A is Noetherian.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 16 / 26
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A, since:
Lemma
N(Ai ) ⊂ V (√JAi )
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J
and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A, since:
Lemma
N(Ai ) ⊂ V (√JAi )
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A, since:
Lemma
N(Ai ) ⊂ V (√JAi )
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A, since:
Lemma
N(Ai ) ⊂ V (√JAi )
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 17 / 26
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A,
since:
Lemma
N(Ai ) ⊂ V (√JAi )
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 17 / 26
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Grauert-Remmert criterion
Non-normal locus N(A) is contained in singular locus Sing(A).
Theorem (Grauert-Remmert)
Let 0 6= J ⊂ A be an ideal with J =√J and
N(A) ⊂ V (J).
Then A is normal iff the inclusion
A ↪→ HomA(J, J)a 7→ a·
is an isomorphism.
=⇒ For J =√
Jac(I ) algorithm terminates with Am = Am+1 = A, since:
Lemma
N(Ai ) ⊂ V (√JAi )
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Local Techniques for Normalization
Theorem (BDLSS, 2011)
SupposeSing(A) = {P1, . . . ,Pr}
andA ⊂ Bi ⊂ A
is the ring given by the normalization algorithm applied to Pi instead of J.Then
(Bi )Pi= APi
(Bi )Q = AQ for all Pi 6= Q ∈ SpecA,
andA = B1 + . . . + Br .
We call Bi the minimal local contribution to A at Pi .
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Local Techniques for Normalization
Theorem (BDLSS, 2011)
SupposeSing(A) = {P1, . . . ,Pr}
andA ⊂ Bi ⊂ A
is the ring given by the normalization algorithm applied to Pi instead of J
.Then
(Bi )Pi= APi
(Bi )Q = AQ for all Pi 6= Q ∈ SpecA,
andA = B1 + . . . + Br .
We call Bi the minimal local contribution to A at Pi .
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Local Techniques for Normalization
Theorem (BDLSS, 2011)
SupposeSing(A) = {P1, . . . ,Pr}
andA ⊂ Bi ⊂ A
is the ring given by the normalization algorithm applied to Pi instead of J.Then
(Bi )Pi= APi
(Bi )Q = AQ for all Pi 6= Q ∈ SpecA,
andA = B1 + . . . + Br .
We call Bi the minimal local contribution to A at Pi .
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 18 / 26
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Local Techniques for Normalization
Theorem (BDLSS, 2011)
SupposeSing(A) = {P1, . . . ,Pr}
andA ⊂ Bi ⊂ A
is the ring given by the normalization algorithm applied to Pi instead of J.Then
(Bi )Pi= APi
(Bi )Q = AQ for all Pi 6= Q ∈ SpecA,
andA = B1 + . . . + Br .
We call Bi the minimal local contribution to A at Pi .
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 18 / 26
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Adjoint ideals
Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ→ Γnormalization map, I (Γ) $ I ⊂ k [x0, ..., xr ] saturated homogeneous ideal.
Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S/I ). Then
0→ IOΓ → π∗(IOΓ)→ F → 0gives for m� 0 linear maps
0→ Im/I (Γ)m$m→ H0
(Γ,OΓ (mH − ∆(I ))
)→ H0 (Γ,F )→ 0
Definition
I is an adjoint ideal of Γ if $m surjective for m� 0.
h0 (Γ,F ) = ∑P∈Sing(Γ) `(IPOΓ,P/IP) =⇒
Theorem
I adjoint ⇐⇒ IPOΓ,P = IP for all P ∈ Sing(Γ).
Conductor is largest ideal with this property.
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Adjoint ideals
Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ→ Γnormalization map, I (Γ) $ I ⊂ k [x0, ..., xr ] saturated homogeneous ideal.Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S/I ).
Then
0→ IOΓ → π∗(IOΓ)→ F → 0gives for m� 0 linear maps
0→ Im/I (Γ)m$m→ H0
(Γ,OΓ (mH − ∆(I ))
)→ H0 (Γ,F )→ 0
Definition
I is an adjoint ideal of Γ if $m surjective for m� 0.
h0 (Γ,F ) = ∑P∈Sing(Γ) `(IPOΓ,P/IP) =⇒
Theorem
I adjoint ⇐⇒ IPOΓ,P = IP for all P ∈ Sing(Γ).
Conductor is largest ideal with this property.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 19 / 26
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Adjoint ideals
Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ→ Γnormalization map, I (Γ) $ I ⊂ k [x0, ..., xr ] saturated homogeneous ideal.Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S/I ). Then
0→ IOΓ → π∗(IOΓ)→ F → 0
gives for m� 0 linear maps
0→ Im/I (Γ)m$m→ H0
(Γ,OΓ (mH − ∆(I ))
)→ H0 (Γ,F )→ 0
Definition
I is an adjoint ideal of Γ if $m surjective for m� 0.
h0 (Γ,F ) = ∑P∈Sing(Γ) `(IPOΓ,P/IP) =⇒
Theorem
I adjoint ⇐⇒ IPOΓ,P = IP for all P ∈ Sing(Γ).
Conductor is largest ideal with this property.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 19 / 26
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Adjoint ideals
Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ→ Γnormalization map, I (Γ) $ I ⊂ k [x0, ..., xr ] saturated homogeneous ideal.Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S/I ). Then
0→ IOΓ → π∗(IOΓ)→ F → 0gives for m� 0 linear maps
0→ Im/I (Γ)m$m→ H0
(Γ,OΓ (mH − ∆(I ))
)→ H0 (Γ,F )→ 0
Definition
I is an adjoint ideal of Γ if $m surjective for m� 0.
h0 (Γ,F ) = ∑P∈Sing(Γ) `(IPOΓ,P/IP) =⇒
Theorem
I adjoint ⇐⇒ IPOΓ,P = IP for all P ∈ Sing(Γ).
Conductor is largest ideal with this property.
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 19 / 26
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Adjoint ideals
Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ→ Γnormalization map, I (Γ) $ I ⊂ k [x0, ..., xr ] saturated homogeneous ideal.Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S/I ). Then
0→ IOΓ → π∗(IOΓ)→ F → 0gives for m� 0 linear maps
0→ Im/I (Γ)m$m→ H0
(Γ,OΓ (mH − ∆(I ))
)→ H0 (Γ,F )→ 0
Definition
I is an adjoint ideal of Γ if $m surjective for m� 0.
h0 (Γ,F ) = ∑P∈Sing(Γ) `(IPOΓ,P/IP) =⇒
Theorem
I adjoint ⇐⇒ IPOΓ,P = IP for all P ∈ Sing(Γ).
Conductor is largest ideal with this property.Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 19 / 26
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Adjoint ideals
Definition
Gorenstein adjoint ideal is the unique largest homogeneous idealG ⊂ K [x0, . . . , xr ] with
GP = COΓ,P for all P ∈ Sing(Γ).
Applications:
Example
If Γ is plane curve of degree n, then Gn−3 cuts out canonical linear series.
Example
If Γ is plane rational of degree n then Gn−2 maps Γ to rational normalcurve of degree n− 2 in Pn−2.
Example
Brill-Noether-Algorithm for computing Riemann-Roch spaces.
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Adjoint ideals
Definition
Gorenstein adjoint ideal is the unique largest homogeneous idealG ⊂ K [x0, . . . , xr ] with
GP = COΓ,P for all P ∈ Sing(Γ).
Applications:
Example
If Γ is plane curve of degree n, then Gn−3 cuts out canonical linear series.
Example
If Γ is plane rational of degree n then Gn−2 maps Γ to rational normalcurve of degree n− 2 in Pn−2.
Example
Brill-Noether-Algorithm for computing Riemann-Roch spaces.
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Adjoint ideals
Definition
Gorenstein adjoint ideal is the unique largest homogeneous idealG ⊂ K [x0, . . . , xr ] with
GP = COΓ,P for all P ∈ Sing(Γ).
Applications:
Example
If Γ is plane curve of degree n, then Gn−3 cuts out canonical linear series.
Example
If Γ is plane rational of degree n then Gn−2 maps Γ to rational normalcurve of degree n− 2 in Pn−2.
Example
Brill-Noether-Algorithm for computing Riemann-Roch spaces.
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Adjoint ideals
Definition
Gorenstein adjoint ideal is the unique largest homogeneous idealG ⊂ K [x0, . . . , xr ] with
GP = COΓ,P for all P ∈ Sing(Γ).
Applications:
Example
If Γ is plane curve of degree n, then Gn−3 cuts out canonical linear series.
Example
If Γ is plane rational of degree n then Gn−2 maps Γ to rational normalcurve of degree n− 2 in Pn−2.
Example
Brill-Noether-Algorithm for computing Riemann-Roch spaces.
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Example
Minimal generators of G for rational curve of degree 5:
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Example
Minimal generators of G for rational curve of degree 5:
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Example
Minimal generators of G for rational curve of degree 5:
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Example
Minimal generators of G for rational curve of degree 5:
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Example
Minimal generators of G for rational curve of degree 5:
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Example
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Local-to-global algorithm
Definition
The local adjoint ideal of Γ at P ∈ Sing Γ is the largest homogeneousideal G(P) ⊂ k [x0, . . . , xr ] with
G(P)P = COΓ,P
Lemma (BDLP, 2015)
G =⋂
P∈Sing ΓG(P)
The G(P) can be computed in parallel via normalization.
Algorithm (BDLP, 2015)
If 1dU is the minimal local contribution at P then
G(P) = (d : U)h
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Local-to-global algorithm
Definition
The local adjoint ideal of Γ at P ∈ Sing Γ is the largest homogeneousideal G(P) ⊂ k [x0, . . . , xr ] with
G(P)P = COΓ,P
Lemma (BDLP, 2015)
G =⋂
P∈Sing ΓG(P)
The G(P) can be computed in parallel via normalization.
Algorithm (BDLP, 2015)
If 1dU is the minimal local contribution at P then
G(P) = (d : U)h
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Local-to-global algorithm
Definition
The local adjoint ideal of Γ at P ∈ Sing Γ is the largest homogeneousideal G(P) ⊂ k [x0, . . . , xr ] with
G(P)P = COΓ,P
Lemma (BDLP, 2015)
G =⋂
P∈Sing ΓG(P)
The G(P) can be computed in parallel via normalization.
Algorithm (BDLP, 2015)
If 1dU is the minimal local contribution at P then
G(P) = (d : U)h
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Special types of singularities
If Γ ⊂ P2 has a singularity of type An at P = (0 : 0 : 1), then given by
f = T 2 +W n+1 with T ,W ∈ C[[x , y ]].
Compute Tj = T +O(j + 1) inductively.
Lemma
If P = (0, 0) is of type An and s =⌊n+12
⌋, then
G(P) = 〈x s , Ts−1, y s〉h ⊂ C[x , y , z ]
Similar results for Dn, En and other singularities in Arnold’s list.
Example
f = x4 − y2 + x5 with A3 singularity. Then G(P) =⟨x2, y
⟩.
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Special types of singularities
If Γ ⊂ P2 has a singularity of type An at P = (0 : 0 : 1), then given by
f = T 2 +W n+1 with T ,W ∈ C[[x , y ]].
Compute Tj = T +O(j + 1) inductively.
Lemma
If P = (0, 0) is of type An and s =⌊n+12
⌋, then
G(P) = 〈x s , Ts−1, y s〉h ⊂ C[x , y , z ]
Similar results for Dn, En and other singularities in Arnold’s list.
Example
f = x4 − y2 + x5 with A3 singularity. Then G(P) =⟨x2, y
⟩.
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Special types of singularities
If Γ ⊂ P2 has a singularity of type An at P = (0 : 0 : 1), then given by
f = T 2 +W n+1 with T ,W ∈ C[[x , y ]].
Compute Tj = T +O(j + 1) inductively.
Lemma
If P = (0, 0) is of type An and s =⌊n+12
⌋, then
G(P) = 〈x s , Ts−1, y s〉h ⊂ C[x , y , z ]
Similar results for Dn, En and other singularities in Arnold’s list.
Example
f = x4 − y2 + x5 with A3 singularity. Then G(P) =⟨x2, y
⟩.
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Special types of singularities
If Γ ⊂ P2 has a singularity of type An at P = (0 : 0 : 1), then given by
f = T 2 +W n+1 with T ,W ∈ C[[x , y ]].
Compute Tj = T +O(j + 1) inductively.
Lemma
If P = (0, 0) is of type An and s =⌊n+12
⌋, then
G(P) = 〈x s , Ts−1, y s〉h ⊂ C[x , y , z ]
Similar results for Dn, En and other singularities in Arnold’s list.
Example
f = x4 − y2 + x5 with A3 singularity. Then G(P) =⟨x2, y
⟩.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.
Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.
Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Modular version of the algorithm
Applying the general modular strategy gives two-fold parallel algorithm.Use primes p such that algorithm is applicable to Γp defined by I (Γ)p.Verification?
Theorem (Arbarello, Ciliberto, 1983, Chiarli, 1984)
Let I (Γ) $ I ⊂ k [x0, . . . , xr ] be saturated homogeneous. Then
deg ∆(I ) ≤ deg I + δ(Γ),
and I is an adjoint ideal of Γ iff
deg ∆(I ) = deg I + δ(Γ).
Theorem (BDLP, 2015, corollary to Lipman, 2006)
δ(Γ) ≤ δ(Γp)
and δ-constant flat family admits a simultaneous normalization.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))
2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)
4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Verification
d(g) = deg(divisor cut out by g away from Sing(Γ)).
Theorem (BDLP, 2015)
Let I ⊂ k [x0, . . . , xr ] be saturated homogeneous with I (Γ) $ I andsuppose G is reduced Grobner basis of I . If p is a prime and g ∈ I ishomogeneous of degree m such that
1 LM(I (Γp)) = LM(I (Γ))2 Gp = G (p) is reduced Grobner basis of an adjoint ideal of Γp
3 d(gp) = (deg Γ) ·m− deg Ip − δ(Γ)4 |mH − ∆(Ip)| is non-special
thendeg ∆(I ) = deg ∆(Ip) = (deg Γ) ·m− d(gp)
δ(Γ) = δ(Γp)
and I is an adjoint ideal.
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Timings in Singular
Plane curve fn of degree n with (n−12 ) singularities of type A1.
par
alle
l
prob
ablis
itic
f5 f6 f7
locNormal 2.1 56 -Maple-IB 5.1 47 318
LA 98 4400 -IQ 1.3 54 3800locIQ � 1.3 (1) 54 (1) 3800 (1)ADE � .18 (1) 1.2 (1) 49 (1)
modLocIQ 6.4 [33] 19 [53] 150 [75]� 6.2 [33] 18 [53] 104 [75]
� .36 (74) 1.6 (153) 51 (230)� � .21 (74) 0.48 (153) 5.2 (230)
[primes] (cores)
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Timings in Singular
Plane curve fn of degree n with (n−12 ) singularities of type A1.
par
alle
l
prob
ablis
itic
f5 f6 f7
locNormal 2.1 56 -Maple-IB 5.1 47 318
LA 98 4400 -IQ 1.3 54 3800locIQ � 1.3 (1) 54 (1) 3800 (1)ADE � .18 (1) 1.2 (1) 49 (1)
modLocIQ 6.4 [33] 19 [53] 150 [75]� 6.2 [33] 18 [53] 104 [75]
� .36 (74) 1.6 (153) 51 (230)� � .21 (74) 0.48 (153) 5.2 (230)
[primes] (cores)Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 26 / 26
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Timings in Singular
Plane curve fn,d of degree d with one singularity of type Dn.Curves h1, h2 of degree 20 and 28 in P5.
par
alle
l
prob
ablis
itic
f50,500 f400,500 h1 h2
locNormal .67 4.9 21 -Maple-IB 1830 - N/A N/A
LA - - N/A N/AIQ .67 5.0 30 -locIQ � .67 (1) 5.0 (1) 7.5 (6) -ADE � .58 (1) 5.0 (1) N/A N/AmodLocIQ � 1.5 [2] 24 [2] 27 [3] 2600 [5]
� � .77 (2) 17 (2) 4.0 [27] 59 (69)
[primes] (cores)
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 26 / 26
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Timings in Singular
Plane curve fn,d of degree d with one singularity of type Dn.Curves h1, h2 of degree 20 and 28 in P5.
par
alle
l
prob
ablis
itic
f50,500 f400,500 h1 h2
locNormal .67 4.9 21 -Maple-IB 1830 - N/A N/A
LA - - N/A N/AIQ .67 5.0 30 -locIQ � .67 (1) 5.0 (1) 7.5 (6) -ADE � .58 (1) 5.0 (1) N/A N/AmodLocIQ � 1.5 [2] 24 [2] 27 [3] 2600 [5]
� � .77 (2) 17 (2) 4.0 [27] 59 (69)
[primes] (cores)
Janko Boehm (TU-KL) Modular Computations in Algebraic Geometry 01 October 2015 26 / 26
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References
J. Boehm, W. Decker, C. Fieker, G. Pfister. The use of bad primes inrational reconstruction, Math. Comp. 84 (2015).
J. Boehm, W. Decker, S. Laplagne, G. Pfister, A. Steenpaß, S. Steidel.Parallel algorithms for normalization, J. Symb. Comp. 51 (2013).
J. Boehm, W. Decker, G. Pfister, S. Laplagne. Local to global algo-rithms for the Gorenstein adjoint ideal of a curve, arXiv:1505.05040.
J. Boehm, W. Decker, G. Pfister, S. Laplagne. adjointideal.lib. ASingular 4 library for computing adjoint ideals, Singular distribution.
P. Kornerup, R. T. Gregory, Mapping integers and Hensel codes ontoFarey fractions, BIT 23 (1983).
E. Arnold, Modular algorithms for computing Grobner bases, J. Symb.Comp. 35 (2003).
G.-M. Greuel, S. Laplagne, S. Seelisch, Normalization of rings, J.Symb. Comp. (2010).
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