Improved effective Łojasiewicz inequality and applications
Let $\mathrm {R}$ be a real closed field. Given a closed and bounded semialgebraic set $A \subset \mathrm {R}^n$ and semialgebraic continuous functions $f,g:A \rightarrow \mathrm {R}$ such that $f^{-1}(0) \subset g^{-1}(0)$ , there exist an integer $N> 0$ and...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000665/type/journal_article |
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| Summary: | Let
$\mathrm {R}$
be a real closed field. Given a closed and bounded semialgebraic set
$A \subset \mathrm {R}^n$
and semialgebraic continuous functions
$f,g:A \rightarrow \mathrm {R}$
such that
$f^{-1}(0) \subset g^{-1}(0)$
, there exist an integer
$N> 0$
and
$c \in \mathrm {R}$
such that the inequality (Łojasiewicz inequality)
$|g(x)|^N \le c \cdot |f(x)|$
holds for all
$x \in A$
. In this paper, we consider the case when A is defined by a quantifier-free formula with atoms of the form
$P = 0, P>0, P \in \mathcal {P}$
for some finite subset of polynomials
$\mathcal {P} \subset \mathrm {R}[X_1,\ldots ,X_n]_{\leq d}$
, and the graphs of
$f,g$
are also defined by quantifier-free formulas with atoms of the form
$Q = 0, Q>0, Q \in \mathcal {Q}$
, for some finite set
$\mathcal {Q} \subset \mathrm {R}[X_1,\ldots ,X_n,Y]_{\leq d}$
. We prove that the Łojasiewicz exponent in this case is bounded by
$(8 d)^{2(n+7)}$
. Our bound depends on d and n but is independent of the combinatorial parameters, namely the cardinalities of
$\mathcal {P}$
and
$\mathcal {Q}$
. The previous best-known upper bound in this generality appeared in P. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991) and depended on the sum of degrees of the polynomials defining
$A,f,g$
and thus implicitly on the cardinalities of
$\mathcal {P}$
and
$\mathcal {Q}$
. As a consequence, we improve the current best error bounds for polynomial systems under some conditions. Finally, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)). |
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| ISSN: | 2050-5094 |