monomial hyperovals in desarguesian planes

Monomial HyperovalsAttempts at Classifying Monomial Hyperovals

Monomial Hyperovals in Desarguesian Planes

Timothy [email protected]

University of Colorado Denver

March 29, 2009

Timothy Vis [email protected] Monomial Hyperovals in Desarguesian Planes


IntroductionClassificationEquivalent Forms

Hyperovals

Definition

In a projective plane of even order q, a hyperoval is a set of q + 2points, no three collinear.

Theorem

In the plane PG(

2, 2h)

, every hyperoval is projectively equivalent

to the set of points

D (f ) ={

(1, x , f (x)) |x ∈ GF(

2h)}

∪ {(0, 1, 0) , (1, 0, 0)} .

In this description of a hyperoval, f is called an o-polynomial.




Monomial Hyperovals

If a hyperoval has a monomial o-polynomial, it is called amonomial hyperoval. We have two reasons for studying these:

1 Classifying all hyperovals is too hard, but classifying monomialhyperovals might be within reach, and

2 Monomial hyperovals have nice groups.

Theorem (O’Keefe, Penttila 1994)

A q − 1 arc with a transitive homography stabilizer is a monomial

(q − 1)-arc or q = 212e+2.

We write D (k) for D(

xk)

.




Examples

The currently known monomial hyperovals are in four families:

Translation hyperovals: D(

2i)

, (i , h) = 1

Segre hyperovals: D (6), h odd

Glynn II hyperovals: D (σ + γ), γ4 = σ

2 = 2, h odd

Glynn III hyperovals: D (3σ + 4), σ2 = 2, h odd.




1-Bit and 2-Bit Monomial Hyperovals

Theorem (Segre 1957)

If D(

2i0)

is a hyperoval, it is a translation hyperoval.

Theorem (Cherowitzo-Storme 1998)

If D(

2i0 + 2i1)

is a hyperoval, it is a translation hyperoval, Segre

hyperoval, or Glynn II hyperoval.




Monomial Hyperovals in Small Planes

Theorem (Glynn 1989)

For h ≤ 28, the only monomial hyperovals are the translation

hyperovals, Segre hyperovals, and Glynn hyperovals.

More recently, unpublished work has extended the upper bound onh to something around 50.

Conjecture

The only monomial hyperovals are the translation hyperovals,

Segre hyperovals, and Glynn hyperovals.




The 3-Bit Classification

Theorem (V.)

If D(

2i0 + 2i1 + 2i2)

is a hyperoval in PG(

2, 2h)

, it is a translation

hyperoval, Segre hyperoval, or Glynn hyperoval.

Proof.

A torturous cases argument that currently stretches 60 pages,

making extensive use of Glynn’s Criterion.




Equivalent o-Monomials

By permuting the points (0, 0, 1), (0, 1, 0), and (1, 0, 0), (orequivalently permuting the coordinates) we obtain six projectivelyequivalent o-monomials for a given k:

e k

(012) 1 −1

k

(021)1

1 − k

(01) 1 − k

(02)k

k − 1

(12)1

k




Observation

Every known monomial hyperoval has at least one representationin at most three bits. Thus, it might make sense to explore whatthe other forms of anything in at most three bits are.



Equivalent FormsTranslation HyperovalsPermutation PolynomialDerivation

Equivalent Forms for 1-Bit

For a single bit 2i the easy forms are:

k =2i

1

k=2h−i

1 − k =

h−1∑

c=i

αc

1 −1

k=

h−1∑

c=h−i

αc




1 − k in General

In general, 1 − k is easy to determine:

Write k = 2i +∑h−1

c=i+1 ac2c with ac ∈ {0, 1}

Let bc = 0 if ac = 1 and let bc = 1 when ac = 0

Then 1 − k = 2i +∑h−1

c=i+1 bc2c .

Example

k =001001101100010000

1 − k =110110010011110000

Unfortunately, no general formula exists for computing inverses.




11−k

for 1-Bit

For a single bit, 1 − k is just a string of consecutive bits.

Theorem (V.)

Let k = 2i . Then 11−k

=∑m

c=1 2c(h−i), where m (h − i) ≡ 1mod h.

Essentially, this leads to an algorithm: place a one in the 2h−i

position and in every position i to the right until the 2 position isreached.




1k

for 2-Bit

We can also determine 1k

for two bits.

Theorem (V.)

If 1(2i+2j )

is defined and (h, j − i) = d, the quotient hd

= 2k + 1 is

odd and

1

(2i + 2j)= 2d−1−i +

k−1∑

l=0

(

d+2∑

m=0

2m+j−i + 2d−1+2(j−i)

)

22l(j−i)−i.




Other Forms

The two forms demonstrated have essentially one parametereach

11−k

for a two bit and 1k

for a three bit have essentially twoparameters each

11−k

for a three bit has essentially three parameters

The remaining forms can be determined using the knownformula for 1 − k.

Given the difficulty with just one parameter, other ideas willprobably be required.




A More Subtle Observation

Consider the 1 − k form for a Glynn III monomial hyperoval:

1 − k =

h−4∑

c=1

αc

α = 2h−1

2

Given this representation, all known monomial hyperovals have arepresentation as

∑mc=1 α

c , where α = 2i , (i , h) = 1.

Translation∑1

c=1 αc

Segre∑2

c=1 αc

α = 2

Glynn II∑2

c=1 αc

α = γ, γ4 = 2

Glynn III∑h−4

c=1 αc

α = 2h−1

2




A Yet More Subtle Observation

For α + α2, notice that if α = γ, we have a Glynn II

hyperoval; if α = γ2 = σ, we have a translation hyperoval; if

α = γ4 = σ

2 = 2, we have a Segre hyperoval.

For∑

h+12

c=1 αc , if α = 1

σ, we have a Glynn II hyperoval; if

α = 12 , we have a translation hyperoval; if α = 1

4 , we have aSegre hyperoval.

In fact, for every such representation of a Glynn hyperoval (ofeither type), replacing α with α

2 yields a translation hyperovaland for every such representation of a Segre hyperoval

replacing α with α12 yields a translation hyperoval.




An Objective

It seems a worthy pursuit to search for some “geometric”meaning to this characteristic of the known monomialhyperovals.

Since the group of a monomial hyperoval acts transitively onthe points off the triangle of reference, a focus on the triangleof reference may be appropriate.




A Related Permutation Polynomial

Consider the line through (1, x , xn) and (1, 1, 1). This line

intersects the line [1, 0, 0] in the point(

0, 1,∑n−1

c=0 xc)

.

(1, x , xn)

(1, 1, 1)

(0, 1, 0)

(0, 0, 1)

(

0, 1,∑n−1

c=0 xc)

[1, 0, 0]

Then∑n−1

c=0 xc must be apermutation polynomial if D (n)is a hyperoval.Conversely, if

∑n−1c=0 xc is a

permutation polynomial, D (n) isa hyperoval.




Another Thought

What if we could find some way of mapping a monomialhyperoval to a related translation hyperoval?

The derivation technique due to Basile and Brutti (1979) maybe the key if we can generalize it.




Derivation

Given a projective plane π that is (P ,P)-transitive for some pointP , and a set of points S intersecting each line through P exactlyonce, let π

′ be the geometry whose points are points of π andwhose lines are lines of π through P and images of S under allelations with center P .

Theorem (V.)

π′ is a projective plane isomorphic to π.

The proof merely observes that the assumption that S be an ovalin Basile and Brutti’s work is not necessary.




Deriving a Desarguesian Plane

In PG (2, q), we may assume P = (0, 0, 1) and then

S = D (f ) \ {P} .

Then derivation is equivalent to the point map σ, where

(1, x , y)σ = (1, x , y − f (x) − x (f (0) − f (1)) + f (0))

(0, 1, y)σ = (0, 1, y − f (0) + f (1))

(0, 0, 1)σ = (0, 0, 1)




Composing Derivations

If we derive with respect to D (f ) and then with respect toD (g), the result is simply a derivation with respect toD (f + g).

Thus, we can map D (f ) to D (g) by deriving with respect toD (f − g).

In particular, we can map a monomial hyperoval D (k) to a(seemingly related) translation hyperoval D (k ′) by deriving

with respect to D(

xk + xk′

)

.

Is there anything noteworthy about D(

xk + xk′

)

?




The End of the Matter...for Today

I don’t know yet.




The End of the Matter...for Today

I don’t know yet.

Thank-you!Questions?


monomial hyperovals in desarguesian planes

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