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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Cambridge University Press
2024-01-01
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| author | Saugata Basu Ali Mohammad-Nezhad |
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| collection | DOAJ |
| description | 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)). |
| format | Article |
| id | doaj-art-56a705d1cddf4ab094082889265cfe76 |
| institution | DOAJ |
| issn | 2050-5094 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-56a705d1cddf4ab094082889265cfe762025-08-20T02:51:31ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.66Improved effective Łojasiewicz inequality and applicationsSaugata Basu0https://orcid.org/0000-0002-2441-0915Ali Mohammad-Nezhad1https://orcid.org/0000-0001-6760-3218Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USADepartment of Statistics and Operations Research, University of North Carolina at Chapel Hill, 318 Hanes Hall, Chapel Hill, NC 27599-3260, USA; E-mail: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)).https://www.cambridge.org/core/product/identifier/S2050509424000665/type/journal_article14P1003C6490C23 |
| spellingShingle | Saugata Basu Ali Mohammad-Nezhad Improved effective Łojasiewicz inequality and applications Forum of Mathematics, Sigma 14P10 03C64 90C23 |
| title | Improved effective Łojasiewicz inequality and applications |
| title_full | Improved effective Łojasiewicz inequality and applications |
| title_fullStr | Improved effective Łojasiewicz inequality and applications |
| title_full_unstemmed | Improved effective Łojasiewicz inequality and applications |
| title_short | Improved effective Łojasiewicz inequality and applications |
| title_sort | improved effective lojasiewicz inequality and applications |
| topic | 14P10 03C64 90C23 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000665/type/journal_article |
| work_keys_str_mv | AT saugatabasu improvedeffectivełojasiewiczinequalityandapplications AT alimohammadnezhad improvedeffectivełojasiewiczinequalityandapplications |