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Proceedings of the American Mathematical Society

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

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Betti numbers of sign conditions
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by Saugata Basu, Richard Pollack and Marie-Françoise Roy PDF
Proc. Amer. Math. Soc. 133 (2005), 965-974 Request permission

Abstract:

Let $\mathrm {R}$ be a real closed field and let ${\mathcal Q}$ and ${\mathcal P}$ be finite subsets of $\mathrm {R}[X_1,\ldots ,X_k]$ such that the set ${\mathcal P}$ has $s$ elements, the algebraic set $Z$ defined by $\bigwedge _{Q \in {\mathcal Q}}Q=0$ has dimension $k’$ and the elements of${\mathcal Q}$ and ${\mathcal P}$ have degree at most $d$. For each $0 \leq i \leq k’,$ we denote the sum of the $i$-th Betti numbers over the realizations of all sign conditions of ${\mathcal P}$ on $Z$ by $b_i({\mathcal P},{\mathcal Q})$. We prove that \[ b_i({\mathcal P},{\mathcal Q}) \le \sum _{j=0}^{k’ - i} {s \choose j} 4^{j} d(2d-1)^{k-1}. \] This generalizes to all the higher Betti numbers the bound ${s \choose k’}O(d)^k$ on $b_0({\mathcal P},{\mathcal Q})$. We also prove, using similar methods, that the sum of the Betti numbers of the intersection of $Z$ with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form $P \geq 0$ or $P\leq 0$ for $P\in {\mathcal P}$, is bounded by \[ \sum _{i = 0}^{k’}\sum _{j = 0}^{k’ - i} {s \choose j} 6^{j} d(2d-1)^{k-1}, \] making the bound $s^{k’} O(d)^k$ more precise.
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Additional Information
  • Saugata Basu
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 351826
  • Email: saugata@math.gatech.edu
  • Richard Pollack
  • Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
  • Email: pollack@cims.nyu.edu
  • Marie-Françoise Roy
  • Affiliation: IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu 35042 Rennes cedex, France
  • Email: mfroy@maths.univ-rennes1.fr
  • Received by editor(s): July 3, 2002
  • Received by editor(s) in revised form: October 10, 2003
  • Published electronically: November 19, 2004
  • Additional Notes: The first author was supported in part by NSF grant CCR-0049070 and an NSF Career Award 0133597.
    The second author was supported in part by NSA grant MDA904-01-1-0057 and NSF grants CCR-9732101 and CCR-0098246.
  • Communicated by: Michael Stillman
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 965-974
  • MSC (2000): Primary 14P10; Secondary 14P25
  • DOI: https://doi.org/10.1090/S0002-9939-04-07629-4
  • MathSciNet review: 2117195