Monomial Hyperovals in Desarguesian Planes

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1 Timothy Vis University of Colorado Denver March 29, 2009

2 Hyperovals Monomial Hyperovals Introduction Classification Definition In a projective plane of even order q, a hyperoval is a set of q + 2 points, no three collinear. Theorem In the plane PG ( 2,2 h), every hyperoval is projectively equivalent to the set of points { ( D (f ) = (1,x,f (x)) x GF 2 h)} {(0,1,0),(1,0,0)}. In this description of a hyperoval, f is called an o-polynomial.

3 Monomial Hyperovals Introduction Classification If a hyperoval has a monomial o-polynomial, it is called a monomial hyperoval. We have two reasons for studying these: 1 Classifying all hyperovals is too hard, but classifying monomial hyperovals 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 = 2 12e+2. We write D (k) for D ( x k).

4 Examples Monomial Hyperovals Introduction Classification The currently known monomial hyperovals are in four families: Translation hyperovals: D ( 2 i), (i,h) = 1 Segre hyperovals: D (6), h odd Glynn II hyperovals: D (σ + γ), γ 4 = σ 2 = 2, h odd Glynn III hyperovals: D (3σ + 4), σ 2 = 2, h odd.

5 Introduction Classification 1-Bit and 2-Bit Monomial Hyperovals Theorem (Segre 1957) If D ( 2 i 0) is a hyperoval, it is a translation hyperoval. Theorem (Cherowitzo-Storme 1998) If D ( 2 i i 1) is a hyperoval, it is a translation hyperoval, Segre hyperoval, or Glynn II hyperoval.

6 Introduction Classification 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 More recently, unpublished work has extended the upper bound on h to something around 50. Conjecture The only monomial hyperovals are the translation hyperovals, Segre hyperovals, and Glynn hyperovals.

7 The 3-Bit Classification Introduction Classification Theorem (V.) If D ( 2 i i i 2) is a hyperoval in PG ( 2,2 h ), 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.

8 Equivalent o-monomials Introduction Classification By permuting the points (0,0,1), (0,1,0), and (1,0,0), (or equivalently permuting the coordinates) we obtain six projectively equivalent 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

9 Observation Introduction Classification Every known monomial hyperoval has at least one representation in at most three bits. Thus, it might make sense to explore what the other forms of anything in at most three bits are.

10 for 1-Bit For a single bit 2 i the easy forms are: k =2 i 1 k =2h i h 1 1 k = c=i 1 1 h 1 k = α c c=h i α c

11 1 k in General In general, 1 k is easy to determine: Write k = 2 i + h 1 c=i+1 a c2 c with a c {0,1} Let b c = 0 if a c = 1 and let b c = 1 when a c = 0 Then 1 k = 2 i + h 1 c=i+1 b c2 c. Example k = k = Unfortunately, no general formula exists for computing inverses.

12 1 1 k Monomial Hyperovals for 1-Bit For a single bit, 1 k is just a string of consecutive bits. Theorem (V.) Let k = 2 i. Then 1 1 k = m c=1 2c(h i), where m (h i) 1 mod h. Essentially, this leads to an algorithm: place a one in the 2 h i position and in every position i to the right until the 2 position is reached.

13 1 k Monomial Hyperovals for 2-Bit We can also determine 1 k for two bits. Theorem (V.) 1 If (2 i +2 j ) is defined and (h,j i) = d, the quotient h d odd and = 2k + 1 is 1 (2 i + 2 j ) = 2d 1 i + k 1 l=0 ( d+2 m=0 2 m+j i + 2 d 1+2(j i) ) 2 2l(j i) i.

14 Other Forms The two forms demonstrated have essentially one parameter each 1 1 k for a two bit and 1 k for a three bit have essentially two parameters each 1 1 k for a three bit has essentially three parameters The remaining forms can be determined using the known formula for 1 k. Given the difficulty with just one parameter, other ideas will probably be required.

15 A More Subtle Observation Consider the 1 k form for a Glynn III monomial hyperoval: h 4 1 k = c=1 α c α = 2 h 1 2 Given this representation, all known monomial hyperovals have a representation as m c=1 αc, where α = 2 i, (i,h) = 1. Translation 1 c=1 αc Segre 2 c=1 αc α = 2 Glynn II 2 c=1 αc α = γ, γ 4 = 2 Glynn III h 4 c=1 αc α = 2 h 1 2

16 A Yet More Subtle Observation For α + α 2, notice that if α = γ, we have a Glynn II hyperoval; if α = γ 2 = σ, we have a translation hyperoval; if α = γ 4 = σ 2 = 2, we have a Segre hyperoval. For h+1 2 c=1 αc, if α = 1 σ, we have a Glynn II hyperoval; if α = 1 2, we have a translation hyperoval; if α = 1 4, we have a Segre hyperoval. In fact, for every such representation of a Glynn hyperoval (of either type), replacing α with α 2 yields a translation hyperoval and for every such representation of a Segre hyperoval replacing α with α 1 2 yields a translation hyperoval.

17 An Objective It seems a worthy pursuit to search for some geometric meaning to this characteristic of the known monomial hyperovals. Since the group of a monomial hyperoval acts transitively on the points off the triangle of reference, a focus on the triangle of reference may be appropriate.

18 A Related Consider the line through (1,x,x n ) and ((1,1,1). This line intersects the line [1,0,0] in the point 0,1, n 1 c=0 ). xc (0,1,0) [1,0,0] ( 0,1, ) n 1 c=0 xc (0,0,1) (1,1,1) (1,x,x n ) Then n 1 c=0 xc must be a permutation polynomial if D (n) is a hyperoval. Conversely, if n 1 c=0 xc is a permutation polynomial, D (n) is a hyperoval.

19 Another Thought What if we could find some way of mapping a monomial hyperoval to a related translation hyperoval? The derivation technique due to Basile and Brutti (1979) may be the key if we can generalize it.

20 Monomial Hyperovals Given a projective plane π that is (P,P)-transitive for some point P, and a set of points S intersecting each line through P exactly once, let π be the geometry whose points are points of π and whose lines are lines of π through P and images of S under all elations with center P. Theorem (V.) π is a projective plane isomorphic to π. The proof merely observes that the assumption that S be an oval in Basile and Brutti s work is not necessary.

21 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)

22 Composing s If we derive with respect to D (f ) and then with respect to D (g), the result is simply a derivation with respect to D (f + g). Thus, we can map D (f ) to D (g) by deriving with respect to D (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 x k + x k ). ( Is there anything noteworthy about D x k + x k )?

23 The End of the Matter...for Today I don t know yet.

24 The End of the Matter...for Today I don t know yet. Thank-you! Questions?

Three-Bit Monomial Hyperovals

Three-Bit Monomial Hyperovals Timothy Vis Timothy.Vis@ucdenver.edu University of Colorado Denver Rocky Mountain Discrete Math Days 2008 Definition of a Hyperoval Definition In a projective plane of order n, a set of n + 1-points, no