On the number of homogeneous nondegenerate p-ary functions of the given degree | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2018. № 41. DOI: 10.17223/20710410/41/1

Let p be a prime number and F = GF(p). Suppose V_n is an n-dimensional vector space over F and e is a basis of Vn. Also, let ф: V_n м F. The function ф is called e-homogeneous if ф(х) = n_V,_e(x) for all x £ V_n, where n_V,_e is an n-variate homogeneous polynomial over F of degree at most p - 1 in each variable and x is the coordinate vector of x with respect to the basis e. The function ф is said to be nondegenerate if deg ф ^ 1 and deg d_v

n \ {0}, where (d_vф)(х) = ф(х + v) - ф(х) for all v,x £ Vn. This notion was introduced by O. A. Logachev, A. A. Sal'nikov, and V. V. Yashchenko in the case when p = 2. Our main results are as follows. First, we obtain a formula for the number HN_p(n,d) of e-homogeneous nondegenerate functions ф: V_n м F of degree d (this number does not depend on e). Namely, if n ^ 1 and d £ {1,...,n(p - 1)}, then HN_p(n, d) = n (-1)^kp^(2)+{n-fc} P M = £ (-1)^|S|p^{CT(S)-|S|+{n-}i^S|} P, where |is the fc=0 L^kJ p SC{1.....n} l^dJ p is the Gaussian binomial coefficient, ^p and a(S) is the sum of all elements of S. The proof of this formula is based on the Mobius inversion. Previously, only formulas for HN_p(n, 2) were known; unlike our formula, their forms depend on the parities of p and n. Second, we prove that HN_p(n, d) ^ p^{d}p - 1 - (pⁿ - 1) ( p^{ d }p - 1 ) /(p-1) for any d ^ 1 and n ^ d/(p-1). ⁿ Using this bound, we obtain that if d ^ 3, then HN_p(n,d) ~ p^{d}p as n м ж. For p = 2 the last two statements were proved by Yu. V. Kuznetsov. The proofs of our main results use a Jennings basis of the group algebra FG_n, where G_n is an elementary abelian p-group of rank n.