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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Zeros of polynomials over finite principal ideal rings

Authors: Murray Marshall and Garry Ramage
Journal: Proc. Amer. Math. Soc. 49 (1975), 35-38
MathSciNet review: 0360541
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Abstract: Let $ R$ be a finite commutative ring with identity. For $ f \in R[{X_1}, \cdots ,{X_n}]$ denote by $ N(f)$ the number of zeros of $ f$ in $ {R^{(n)}}$. For integers $ n,d \geq 1$ denote by $ {A_{n,d}}$ the greatest common divisor of the integers $ N(f);f \in R[{X_1}, \cdots ,{X_n}],\deg f = d$. J. Ax has shown that if $ R$ is a field, then $ {A_{n,d}} = \vert R{\vert^\alpha }$ where $ \alpha $ is the integer satisfying $ \alpha < n/d \leq \alpha + 1$. In this paper, $ {A_{n,d}}$ is computed in the case that $ R$ is a principal ideal ring.

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Article copyright: © Copyright 1975 American Mathematical Society

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