coprime polynomial pairs, hankel matrices, and splitting...

Coprime polynomial pairs, Hankel matrices, andsplitting subspaces

Sudhir R. Ghorpade

Department of MathematicsIndian Institute of Technology Bombay

Powai, Mumbai 400076, Indiahttp://www.math.iitb.ac.in/∼srg/

CanaDAM-2011Victoria, BC, Canada

June 1, 2011

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

References and Names of collaborators

This talk corresponds mainly to the following two papers:

Mario Garcia-Armas, Sudhir R. Ghorpade, and Samrith Ram.Relatively prime polynomials and nonsingular Hankel matricesover finite fields, Journal of Combinatorial Theory, Series A,Vol. 118, No. 3 (2011), pp. 819-828.

Sudhir R. Ghorpade, and Samrith Ram. Block companionSinger cycles, primitive recursive vector sequences, andcoprime polynomial pairs over finite fields, Finite Fields andTheir Applications, (2011), doi:10.1016/j.ffa.2011.02.008.

A precursor to the latter was the following paper:

Sudhir R. Ghorpade, Sartaj Ul Hasan and Meena Kumari,Primitive polynomials, Singer cycles, and word-oriented linearfeedback shift registers, Designs, Codes and Cryptography,Vol. 58, No. 2 (2011), pp. 123-134.

All these (and more!) are available at:http://www.math.iitb.ac.in/∼srg/Papers.html


Some Questions

What is the probability that two polynomials in Fq[X ] ofdegree n ≥ 1 are relatively prime?

What is the probability that a n × n Hankel matrix over Fq isnonsingular? Similarly, what is the probability that a n × nToeplitz matrix over Fq is nonsingular?[Recall: A = (aij) is Hankel if aij = ars whenever i + j = r + s,and it is Toeplitz if aij = ars whenever i − j = r − s.]

(Niederreiter, 1995) Given σ such that Fqmn = Fq(σ), what isthe number of σ-splitting subspaces of Fqmn of dimension m?[Recall: An m-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]


Coprime Polynomial Pairs

Thanks to Knuth [ACP-2 (§4.6.1, Ex. 5), 1969] and more recently,Corteel, Savage, Wilf, Zeilberger (and Zagier) [JCT-A, 1998], weknow that the probability that two monic polynomials of degreen ≥ 1 over Fq, chosen independently and uniformly at random, arerelatively prime is

1− 1

q.

A bijective “explanation” was given by Reifegerste (2000) in thecase q = 2. The case of arbitrary q was “explained” by Benjaminand Bennett (2007) by constructing an explicit surjective map

{(f , g) : f , g ∈ Fq[X ] monic, deg n, coprime}→ {(f , g) : f , g monic, deg n, non-coprime}

such that the cardinality of each fiber is q − 1.


Side Remark: Connection with Riemann Zeta Function

It is an elementary and well-known that the probability of twointegers to be relatively prime is

1

ζ(2)=

6

π2.

To note that the result for polynomials fits in with this, recall thatthe Weil and Riemann zeta functions of any variety V over Fq are

ZV (T ) := exp

( ∞∑n=1

|V (Fqn)| T n

n

)and ζV (s) := ZV

(q−s).

Now Fq[X ]←→ A1Fq

and

ZA1(T ) = exp

( ∞∑n=1

qn T n

n

)= exp

(log(1− qT )−1

)=

1

1− qT

=⇒ ζA1(2) =1

1− q(q−2)and

1

ζA1(2)= 1− 1

q.


Nonsingular Hankel Matrices

Daykin [Crelle, 1960] was perhaps the first to prove that thenumber of n × n nonsingular Hankel matrices over Fq isq2n−2(q − 1). In particular, the probability that a Hankel matrixover Fq is nonsingular is

1− 1

q.

[More generally, Daykin has formulas for the number of m × nHankel matrices over Fq of a given rank r ≤ min{m, n}. ]

Recently, Daykin’s formula has been (re)proved [usingsubresultants and such] by Kaltofen and Lobo (1996) in thecontext of counting n× n nonsingular Toeplitz matrices over Fq. Asimpler and more direct proof can be found in our paper in JCT-A118 (2011), 819828. The case of Hankel matrices over Fq of agiven rank is also discussed there.


Coprime Polynomial Pairs and Nonsingular Hankel Matrices

Curious Fact: The probability for a polynomial pair to be coprimeand for a Hankel matrix to be nonsingular is the same! In fact,

|CPPn(Fq)| = q2n

(1− 1

q

)= q2n−1(q − 1),

and

|HGLn(Fq)| = q2n−1(

1− 1

q

)= q2n−2(q − 1),

where CPPn(Fq) denotes the set of all ordered pairs of coprimemonic polynomials over Fq of degree n and HGLn(Fq) denotes theset of all n × n nonsingular Hankel matrices with entries in Fq.

Question: Can we “explain” this fact? In other words, can we givea combinatorial proof that |CPPn(Fq)| = q |HGLn(Fq)|?


An explicit surjection

Let F be any field and n be any positive integer.

Theorem

There is a surjective map σ : CPPn(F )→ HGLn(F ) such thatfor any A ∈ HGLn(F ), the fiber σ−1 ({A}) is in one-to-onecorrespondence with F . In particular, |CPPn(Fq)| = q |HGLn(Fq)| .

Sketch of Proof: A key idea is to consider the Bezoutian. Recallthat the Bezoutian (matrix) of u, v ∈ F [X ] of degree ≤ n is then × n matrix Bn(u, v) = (bij) determined by the equation

u(X )v(Y )− v(X )u(Y )

X − Y=

n∑i ,j=1

bijXi−1Y j−1.

Fact: Assume that deg u = n and deg v ≤ n. Then

Bn(u, v) is nonsingular ⇐⇒ u and v are coprime


Illustration of the Bezoutian

Example: Suppose u(X ) = u0 + u1X + · · ·+ unX n withu0, u1, . . . , un ∈ F and v(X ) = 1. Then

u(X )− u(Y )

X − Y=

n∑k=1

ukX k − Y k

X − Y=

n∑k=1

uk

k∑i=1

X i−1 Y k−i

=n∑

i ,j=1

ui+j−1X i−1 Y j−1,

where, by convention, uk := 0 for k > n. Thus

Bn(u, 1) =

u1 u2 · · · un−1 un

u2 u3 · · · un 0... . . . ...

un−1 un · · · 0 0un 0 · · · 0 0

In particular, if deg u = n, i.e., if un 6= 0, then u and v are coprime,and moreover Bn(u, v) is nonsingular.


Pade pairs and Hermite pairs

Define the set of Pade pairs by

Pn(F ) :={

(u, v) ∈ F [X ]2 : u monic, deg u = n, and deg v < n},

and the set of Hermite pairs by

HPn(F ) := {(u, v) ∈ Pn(F ) : u and v are coprime} .

Observe that CPPn(F ) is in bijection with HPn(F ) [Proof: Themap given by (f , g) 7→ (f , g − f ) does the job.]. Next, observe thatfor any (u, v) ∈ Pn(F ), there are unique ai ∈ F , i ≥ 1, such that

v(X )

u(X )=∞∑i=1

aiX i.

Define Hn(u, v) to be the n × n Hankel matrix (ai+j−1).


Completion of the proof

Lemma (Barnett factorization)

For any (u, v) ∈ Pn(F ),

Bn(u, v) = Bn(u, 1)Hn(u, v)Bn(u, 1).

Thanks to the above results, (u, v) 7−→ Hn(u, v) gives a mapHPn(F )→ HGLn(F ). With some effort, we can show that it issurjective and that the fibre is in bijection with F . Combining thiswith the bijection CPPn(F )→ HPn(F ), we obtain the desiredresult.

Remark: One can replace HGLn(F ) by the set TGLn(F ) of n × nnonsingular Toeplitz matrices over Fq. Indeed if E denotes then × n matrix with 1 on the antidiagonal and 0 elsewhere, then E isnonsingular and the map given by A 7→ AE gives a bijectionbetween TGLn(F ) and HGLn(F ).


Splitting Subspaces

Let m, n be positive integers and σ ∈ Fqmn . Recall that anm-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]

Denote by S(σ,m, n; q) the number of m-dimensional σ-splittingsubspaces of Fqmn .

Note: For an arbitrary σ ∈ Fqmn , there may not be any σ-splittingsubspace; for example, this happens if σ ∈ Fq and n > 1. However,if σ ∈ Fqmn satisfies Fqmn = Fq(σ), then a σ-splitting subspaceexists and, in fact, S(σ,m, n; q) ≥ (qmn − 1)/(qm − 1).

Examples: If min{m, n} = 1 and if σ ∈ Fqmn is such thatFqmn = Fq(σ), then it is easy to see that

S(σ,m, n; q) =qmn − 1

qm − 1.


Quantitative Formulation of Niederreiter’s Question

Splitting Subspace Conjecture: For any σ such that Fqmn = Fq(σ),

S(σ,m, n; q) =qmn − 1

qm − 1qm(m−1)(n−1).

One can have a simpler formulation using the following notion:

Definition

By a pointed σ-splitting subspace of dimension m we shall mean apair (W , x) where W is an m-dimensional σ-splitting subspace ofFqmn and x ∈W . The element x may be referred to as the basepoint of (W , x).

Pointed Splitting Subspace Conjecture: For any x ∈ F∗qmn and anyσ such that Fqmn = Fq(σ), the number of m-dimensional pointed

σ-splitting subspaces of Fqmn with base point x is qm(m−1)(n−1).Consequently, there is a one-to-one correspondence betweenpointed splitting subspaces and pointed n-tuples of m ×mnilpotent matrices over Fq.


What do we know about the SSC and PSSC?

The case min{m, n} = 1 is easy. The best known general resultseems to be the following.

Theorem

The Splitting Subspace Conjecture as well as the Pointed SplittingSubspace Conjecture holds in the affirmative if m = 2.

We remark that one of the key ingredients in the proof of thistheorem is a result closely related to the question about theprobability for a pair of polynomials in Fq[X ] to be relatively prime.More precisely, we use a result of Benjamin and Bennett about thenumber of coprime pairs of polynomials in Fq[X ] of degree < n.


Bounds and Asymptotics

For any σ ∈ Fqmn , defineN(σ,m, n; q) := S(σ,m, n; q)

∏m−1i=0 (qm − qi ).

Theorem

Let σ ∈ Fqmn be such that Fqmn = Fq(σ).

(q − 2)qmn + 1

(q − 1)qmn(m−1) ≤ N(σ,m, n; q) ≤

m−1∏i=0

(qmn − qi ).

Corollary

Let σ ∈ Fqmn be such that Fqmn = Fq(σ) and let x ∈ F∗qmn . Thenthe number of m-dimensional pointed σ-splitting subspaces of Fqmn

with base point x is is asymptotically equivalent to qm(m−1)(n−1)

as q →∞.


Related Problems

The above conjecture is closely related to:

1 Counting the number of primitive σ-LFSRs of order n overFqm , or equivalently, the number of primitive recursive vectorsequences of order n over Fm

q .2 Counting the number of (m, n)-block companion Singer cycles

over Fq, i.e., the number of mn ×mn block matrices T of thefollowing form which are nonsingular and whose order inGLmn(Fq) is the meximum possible:

T =

0 0 0 . . 0 0 C0

Im 0 0 . . 0 0 C1

. . . . . . . .

. . . . . . . .0 0 0 . . Im 0 Cn−20 0 0 . . 0 Im Cn−1

,

where C0,C1, . . . ,Cn−1 ∈ Mm(Fq) and Im and 0 denote them ×m identity and zero matrix over Fq, respectively.


Thank you for your attention!


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