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June 2013 Pairs of additive forms of degree $p^\tau(p-1)$
Hemar Godinho, Tertuliano C. de Souza Neto
Funct. Approx. Comment. Math. 48(2): 197-211 (June 2013). DOI: 10.7169/facm/2013.48.2.3

Abstract

Let \[ f(x_1,...,x_n)=a_{1}x_{1}^{k}+\cdots+a_{n}x_{n}^{k}\\ g(x_1,...,x_n)=b_{1}x_{1}^{k}+\cdots+b_{n}x_{n}^{k} \] be a pair of additive forms of degree $k=p^{\tau}(p-1)$. We are interested in finding conditions which guarantee the existence of $p$-adic zeros for this pair of forms. A well-known conjecture due to Emil Artin states that the condition $n > 2k^2$ is sufficient. Here we prove that $$n > 2\left(\frac{p}{p-1}\right)k^2-2k$$ is sufficient, provided that $p > 5$ and $\tau \geq \dfrac{p-1}{2}$.

Citation

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Hemar Godinho. Tertuliano C. de Souza Neto. "Pairs of additive forms of degree $p^\tau(p-1)$." Funct. Approx. Comment. Math. 48 (2) 197 - 211, June 2013. https://doi.org/10.7169/facm/2013.48.2.3

Information

Published: June 2013
First available in Project Euclid: 18 June 2013

zbMATH: 1306.11030
MathSciNet: MR3100140
Digital Object Identifier: 10.7169/facm/2013.48.2.3

Subjects:
Primary: 11D72
Secondary: 11P99

Keywords: $p$-adic forms , additive forms , Artin's conjecture

Rights: Copyright © 2013 Adam Mickiewicz University

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Vol.48 • No. 2 • June 2013
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