Point count divisibility for algebraic sets over ℤ/𝕡^{ℓ}ℤ and other finite principal rings

Katz, Daniel

doi:10.1090/S0002-9939-09-10017-5

Point count divisibility for algebraic sets over ${\mathbb {Z}}/p^\ell {\mathbb {Z}}$ and other finite principal rings
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by Daniel J. Katz PDF

Proc. Amer. Math. Soc. 137 (2009), 4065-4075 Request permission

Abstract:

We determine the greatest common divisor of the cardinalities of the algebraic sets generated by collections of polynomials $f_1,\ldots ,f_t$ of specified degrees $d_1,\ldots ,d_t$ in $n$ variables over a finite principal ring $R$. This generalizes the theorems of Ax ($t=1$, $R$ a field), N. M. Katz ($t$ arbitrary, $R$ a field), and Marshall-Ramage ($t=1$, $R$ an arbitrary finite principal ring).

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