Nice Exponent
computed in April 2013
Definition
Let K be a finite field.
Let s be positive integer.
For all y in K*, p(y) denotes the number
of pre-images of y by
f : x -> x^s + ( 1 - x )^s.
that is
p(y) := # { x | x^s + ( 1 - x )^s = y }
An exponent s is said to be nice when
y->p(y) takes at most three values.
Methodology
Using the notion of Zech logarithm it is very easy
to determine the number of nice exponents over small
fields.
Let n the order of K*.
Let t be a primitive root of K. Any non zero element
x of K is a power of t. There exists a uniq residue
j modulo n such that
x = t^j
It is the logarithm of x. By definition the logarithm
of 1+x is called the Zech logarithm of x. Assuming
p is odd, the logarithm
of the image of x by the map f
j * s + zech[ s * ( zech[ x + n / 2 ] - j )
just because
f(x) = x^s ( 1 + ( ( 1 - x ) / x )^s )
For our numerical experiment, given a finite field K:=GF(p,m) of order q,
we proceed in two steps. First, we compute the table of zech logarithm
it is about O( qm). Using this table, the distribution of the images by
f are easy to compute and the total runtime is about O(q^2).
Numerical data
We compute the distribution for all the finite fields
(not prime ) of characteristic less or equal to 31,
having an order less than 2^20.
In the files below, lines like
#field is GF(3,9)
---
13121 : 1 : 3 : 9841 [ 0] 9840 [ 2] 1 [ 3]
means that the exponent 13121 is nice for the field
GF(3,9) : 9841 elements have no preimage,
9840 have two antecedants. Note that N(1,0) = 0 is
not reported in the files.
Conjectures
On the basis of that numerical experiment,
for a field of odd characteristic p, one
may conjecture :
if s is a nice exponent 0 and 2 are multiplicities
of pre-image size. In particular, this claim implies
Hellesth's conjecture concerning Fourier spectra of
power permutations in an hyper-quadratic extension
of GF(p).
Up to equivalence, on a prime field p such that
gcd(p-1,3)=1 the nice exponents are 3 and p-2. For
the other p, nice exponents do not exist.
Philippe Langevin
Last modifications March 23th 2013,
March 31th 2015.