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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Ideal membership in polynomial rings over the integers
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by Matthias Aschenbrenner PDF
J. Amer. Math. Soc. 17 (2004), 407-441 Request permission


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
  • Email:
  • Received by editor(s): May 2, 2003
  • Published electronically: January 15, 2004
  • Additional Notes: Partially supported by the Mathematical Sciences Research Institute
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 407-441
  • MSC (2000): Primary 13P10; Secondary 11C08
  • DOI:
  • MathSciNet review: 2051617