Rocky Mountain Journal of Mathematics

Rational linear spaces on hypersurfaces over quasi-algebraically closed fields

Todd Cochrane, Craig V. Spencer, and Hee-Sung Yang

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Let $k=\mathbb{F}_q(t)$ be the rational function field over $\mathbb{F}_q$ and $f(\mathbf{x}) \in k[x_1, \ldots, x_s]$ be a form of degree~$d$. For $l \in \mathbb{N}$, we establish that whenever \[ s > l + \sum_{w=1}^{d} w^2 \binom{d-w+l-1}{l-1}, \] the projective hypersurface $f(\mathbf{x})=0$ contains a $k$-rational linear space of projective dimension~$l$. We also show that if $s> 1+ d(d+1)(2d+1)/6$, then for any $k$-rational zero $\mathbf{a}$ of $f(\mathbf{x})$ there are infinitely many $s$-tuples $(\varpi_1, \ldots, \varpi_s)$ of monic irreducible polynomials over $k$, with the $\varpi_i$ not all equal, and $f(a_1\varpi_1, \ldots, a_s \varpi_s) =0$. We establish in fact more general results of the above type for systems of forms over $C_i$-fields.

Article information

Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1805-1816.

First available in Project Euclid: 2 February 2015

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Zentralblatt MATH identifier

Primary: 11D72: Equations in many variables [See also 11P55] 12F20: Transcendental extensions 11T55: Arithmetic theory of polynomial rings over finite fields

Diophantine equations function fields quasi-algebraic closure


Cochrane, Todd; Spencer, Craig V.; Yang, Hee-Sung. Rational linear spaces on hypersurfaces over quasi-algebraically closed fields. Rocky Mountain J. Math. 44 (2014), no. 6, 1805--1816. doi:10.1216/RMJ-2014-44-6-1805.

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