有限域上完全对称多项式有很多零点

Complete symmetric polynomials over finite fields have many rational zeros

有限域上多项式的零点计数问题是算术代数几何的核心问题之一.本文考虑有限域Fq上完全对称多项式的零点问题,主要结果如下:设h(x1,..., xk)∈Fq[x1,..., xk]是有限域Fq上一个m次完全对称多项式(k≥3, 1≤m≤q-3),(1)若m为奇数或者q为奇数,则多项式h(x1,..., xk)在Fqk中至少有6qk-3个零点;(2)若k≥4,则多项式h(x1,..., xk)在Fqk中至少有6(q-1)qk-4个零点.作为推论,我们证明了文献Zhang和Wan (2020)中的猜想1.7.

Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry. In this paper, we consider the problem for complete symmetric polynomials. The homogeneous complete symmetric polynomial of degree m in the k-variables {x;, x;,..., x;} is defined as ■ A complete symmetric polynomial of degree m over F;in the k-variables {x;, x;,..., x;} is defined as ■ where a;∈F;and a;≠0. We are interested in counting the number of zeros and the number of zeros with pairwise distinct coordinates of a complete symmetric polynomial, respectively. Let ■ denote the number of F;-rational points on the affine hypersurface defined by h(x;,..., x;) = 0. Let ■ denote the number of F;-rational points on the affine hypersurface defined by h(x;,..., x;) = 0 with the additional condition that the coordinates are distinct. In the paper Zhang and Wan(2020), the authors showed the lower bound N;(h) ≥6q;if q is odd, k≥3 and 1≤m≤k-3 and conjectured N;(h)≥24 q;if q is even, k≥4 and 1≤m≤k-4. The key ingredient in the proof of the lower bound is to prove N;(h(x;, x;, x;))≥6 for odd q, k = 3 and 1≤m≤q-3 which does not hold for even q in general. In this paper, we deal with the even characteristic case. The main new results are the following(suppose F;is a finite field with characteristic 2):(1) Let h(x;, x;, x;) :=∑;a;h;(x;, x;, x;) ∈ F;[x;, x;, x;] be a complete symmetric polynomial of degree m with 1≤m≤q-3. If a;≠0 for some odd e0, then N;(h)≥6.(2) Let h(x;,..., x;) be a complete symmetric polynomial in k≥3 variables over Fq of degree m with 1≤m≤q-3. If m is odd, then N;(h)≥6 q;.(3) Let h(x;,..., x;) be a complete symmetric polynomial in k≥4 variables over Fq of degree m with 1≤m≤q-3. Then N;(h)≥6(q-1)q;.(4) As a consequence, Conjecture 1.7 in the paper [Zhang J, Wan D. Rational points on complete symmetric hypersurfaces over finite fields. Discrete Math, 2020, 343: 112072] is true. That is, for any complete symmetric polynomial h(x;,..., x;) in k≥4 variables over Fq of degree m with 1≤m≤q-4, we have Nq(h)≥24q;.