Complete symmetric polynomials over finite fields have many rational zeros

Abstract

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 $\\{x_1,x_2,\\ldots,x_k\\}$is defined as <disp-formula><tex-math><![CDATA[$$h_m(x_1,x_2,\\ldots,x_k):=\\sum_{1\\leq i_1\\leq i_2\\leq \\cdots \\leq i_m\\leq k}x_{i_1}x_{i_2}\\cdots x_{i_m}.$$]]></tex-math></disp-formula> A complete symmetric polynomial of degree $m$ over $\\f{q}$ in the $k$-variables $\\{x_1,x_2,\\ldots,x_k\\}$ is defined as <disp-formula><tex-math><![CDATA[$$h(x_1,\\ldots, x_k):=\\sum_{e=0}^m a_eh_e(x_1,x_2,\\ldots,x_k),$$]]></tex-math></disp-formula> where $a_e\\in~\\f{q}$ and $a_m\\not=0$. 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 <disp-formula><tex-math><![CDATA[$$N_q(h):= \\#\\{(a_1,\\ldots, a_k)\\in \\f{q}^k\\mid h(a_1,\\ldots, a_k)=0\\}$$]]></tex-math></disp-formula> denote the number of $\\f{q}$-rational points on theaffine hypersurface defined by $h(x_1,\\ldots,~x_k)=0$. Let <disp-formula><tex-math><![CDATA[$$N_q^*(h):=\\# \\{(a_1,\\ldots, a_k)\\in \\f{q}^k\\mid h(a_1,\\ldots, a_k)=0 \\mbox{and}\\, a_i\\not= a_j, \\forall\\, i\\not=j\\}$$]]></tex-math></disp-formula> denote the number of $\\f{q}$-rational points on theaffine hypersurface defined by $h(x_1,\\ldots,~x_k)=0$with the additional condition that the coordinates are distinct. In the paper Zhang and Wan (2020), the authors showed the lower bound $N_q(h)\\geq~6q^{k-3}$ if $q$ is odd, $k\\geq~3$ and $1\\leq~m\\leq~k-3$ and conjectured $N_q(h)\\geq~24q^{k-4}$ if $q$ is even, $k\\geq~4$ and $1\\leq~m\\leq~k-4.$ The key ingredient in the proof of the lower bound is to prove $N_q^*(h(x_1,x_2,x_3))\\geq~6$ for odd $q$, $k=3$ and $1\\leq~m\\leq~q-3$ 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 followings (suppose $\\f{q}$ is a finite field with characteristic $2$). (1) Let $h(x_1,x_2,~x_3):=\\sum_{e=0}^m~a_eh_e(x_1,x_2,x_3)\\in~\\f{q}[x_1,x_2,x_3]$ be a complete symmetric polynomials of degree $m$ with $1\\leq~m\\leq~q-3$. If $a_{e_0}\\neq~0$ for some odd $e_0$, then $N_q^*(h)\\geq~6.$ (2) Let $h(x_1,\\ldots,~x_k)$ be a complete symmetric polynomials in $k\\geq~3$ variables over $\\f{q}$ of any degree $m$ with $1\\leq~m\\leq~q-3$. If $m$ is odd, then $N_q(h)\\geq~6q^{k-3}.$ (3) Let $h(x_1,\\ldots,~x_k)$be a complete symmetric polynomial in $k\\geq~4$ variables over $\\f{q}$ ofdegree $m$ with $1\\leq~m\\leq~q-3$. Then $N_q(h)\\geq~6(q-1)q^{k-4}.$ (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 $h(x_1,\\ldots,~x_k)$ in $k\\geq~4$ variables over $\\f{q}$ ofdegree $m$ with $1\\leq~m\\leq~q-4$, we have$N_q(h)\\geq~24q^{k-4}.$