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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Ideal membership in polynomial rings over the integers

Author: Matthias Aschenbrenner
Journal: J. Amer. Math. Soc. 17 (2004), 407-441
MSC (2000): Primary 13P10; Secondary 11C08
Published electronically: January 15, 2004
MathSciNet review: 2051617
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Abstract: We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials $f_0,f_1,\dots ,f_n\in \mathbb Z[X]$, where $X=(X_1,\dots ,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,\dots ,g_n\in \mathbb Z[X]$ such that $f_0=g_1f_1+\cdots +g_nf_n$? We show that the degree of the polynomials $g_1,\dots ,g_n$ can be bounded by $(2d)^{2^{O(N\log (N+1))}}(h+1)$ where $d$ is the maximum total degree and $h$ the maximum height of the coefficients of $f_0,\dots ,f_n$. Some related questions, primarily concerning linear equations in $R[X]$, where $R$ is the ring of integers of a number field, are also treated.

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Additional Information

Matthias Aschenbrenner
Affiliation: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720; Department of Mathematics, University of California at Berkeley, Evans Hall, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan St. (M/C 249), Chicago, Illinois 60607

Keywords: Ideal membership over the integers, bounds, restricted power series
Received by editor(s): May 2, 2003
Published electronically: January 15, 2004
Additional Notes: Partially supported by the Mathematical Sciences Research Institute
Article copyright: © Copyright 2004 American Mathematical Society