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- W2959707788 abstract "Given finitely many closed points in distinct fibers of a non-degenerate quadric over $mathbb{A}^1_{mathbb{F}_q}$, we ask for conditions under which there is a section passing through the closed points, possibly with higher order (nilpotence) conditions. This could be thought of as a quadratic version of Lagrange interpolation, and it is equivalent to proving strong approximation for non-degenerate quadrics over $mathbb{F}_q[t]$. We show that under mild conditions on the quadratic form $F$ over $mathbb{F}_q[t]$ in $d$ variables, $f,ginmathbb{F}_q[t]$, $boldsymbol{lambda}inmathbb{F}_q[t]^d$, if $dgeq 5$ then for $°fgeq (4+varepsilon)°g+O(1)$ we have a solution $vec{x}inmathbb{F}_q[t]^d$ to $F(vec{x})=f$ such that $vec{x}equivboldsymbol{lambda}bmod g$, where the the big-Oh notation does not depend on $f,g,boldsymbol{lambda}$. For $d=4$, we show the same is true for $°fgeq (6+varepsilon)°g+O(1)$. This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any $k$-regular Morgenstern Ramanujan graphs $G$ is at most $(2+varepsilon)log_{k-1}|G|+O_{varepsilon}(1)$. In contrast to the $d=4$ case, our result is optimal for $dgeq 5$. Along the way, we prove a stationary phase theorem over function fields that is of independent interest." @default.
- W2959707788 created "2019-07-23" @default.
- W2959707788 creator A5009399128 @default.
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- W2959707788 date "2019-07-18" @default.
- W2959707788 modified "2023-09-27" @default.
- W2959707788 title "Sections of quadrics over $mathbb{A}^1_{mathbb{F}_q}$" @default.
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