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- W2218808082 abstract "Let $finBbb F_q[X_1,dots,X_n]$ with $deg f=d>0$ 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 $|Z(f)|equiv 0pmod {q^{lceil n/drceil-1}}$, that is, $nu_p(|Z(f)|)ge m(lceil n/drceil-1)$, where $p=text{char},Bbb F_q$, $q=p^m$, 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, $nu_p(|Z(f)|)=m(lceil n/drceil-1)$. Let $R_q(d,n)$ denote the $q$-ary Reed-Muller code ${finBbb F_q[X_1,dots,X_n]: deg fle d, deg_{X_j}fle q-1, 1le jle n}$, 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) $q=2$, $n/2le dle n-2$, $t=2$; (iii) $q=3^m$, $d=n$, $t=1$; (iv) $q=3$, $nle dle 2n$, $t=1$." @default.
- W2218808082 created "2016-06-24" @default.
- W2218808082 creator A5039501430 @default.
- W2218808082 date "2015-12-15" @default.
- W2218808082 modified "2023-10-16" @default.
- W2218808082 title "Polynomials Meeting Ax's Bound" @default.
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- W2218808082 doi "https://doi.org/10.48550/arxiv.1512.04997" @default.
- W2218808082 hasPublicationYear "2015" @default.
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