A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Polynomials meeting Ax's bound

Volume 176 / 2016

Xiang-dong Hou Acta Arithmetica 176 (2016), 65-80 MSC: Primary 11L05, 11T06; Secondary 94B27. DOI: 10.4064/aa8405-7-2016 Published online: 30 September 2016

Abstract

Let $f\in\Bbb F_q[X_1,\dots,X_n]$ with $\mathop{\rm deg} f=d \gt 0$ and let $Z(f)=\{(x_1,\dots,x_n)\in \Bbb F_q^n: f(x_1,\dots,x_n)=0\}$. Ax’s theorem states that $|Z(f)|\equiv 0\pmod {q^{\lceil n/d\rceil-1}}$, that is, $\nu_p(|Z(f)|)\ge m(\lceil n/d\rceil-1)$, where $p=\mathop{\rm char} \Bbb F_q$, $q=p^m$, and $\nu_p$ is the $p$-adic valuation. In this paper, we determine a condition on the coefficients of $f$ that is necessary and sufficient for $f$ to meet Ax’s bound, that is, $\nu_p(|Z(f)|)=m(\lceil n/d\rceil-1)$. Let $R_q(d,n)$ denote the $q$-ary Reed–Muller code $\{f\in\Bbb F_q[X_1,\dots,X_n]: \mathop{\rm deg} f\le d,\, \mathop{\rm deg}_{X_j}f\le q-1$, $1\le j\le n\}$, and let $N_q(d,n;t)$ be the number of codewords of $R_q(d,n)$ with weight divisible by $p^t$. As applications of the aforementioned result, we find explicit formulas for $N_q(d,n;t)$ in the following cases: (i) $q=2^m$, $n$ even, $d=n/2$, $t=m+1$; (ii) $q=2$, $n/2\le d\le n-2$, $t=2$; (iii) $q=3^m$, $d=n$, $t=1$; (iv) $q=3$, $n\le d\le 2n$, $t=1$.

Authors

  • Xiang-dong HouDepartment of Mathematics and Statistics
    University of South Florida
    Tampa, FL 33620, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image