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Remarks on Growth Rates of Polynomials on Semialgebraic Sets: Remarks on Growth Rates of Polynomials on Semialgebraic Sets: M. Michalska.

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Title: Remarks on Growth Rates of Polynomials on Semialgebraic Sets: Remarks on Growth Rates of Polynomials on Semialgebraic Sets: M. Michalska.
Authors: Michalska, Maria1 (AUTHOR) Maria.Michalska@wmii.uni.lodz.pl
Source: Discrete & Computational Geometry. Apr2025, Vol. 73 Issue 3, p595-617. 23p.
Subjects: Semialgebraic sets, Homogeneous polynomials, Polynomials, Algebra, Polyhedra
Abstract: Fix a semialgebraic set S in R n and a function g : S → R . We address the question on how to determine the power m such that for a polynomial f the function | f | / | g | m is bounded on S. As a consequence, we give explicit formulae for growth rate of a polynomial, i.e., the optimal power m when g = ‖ x ‖ , on a certain type of sets, called weighted tentacles, in terms of their support. To this aim, we study algebras of bounded polynomials for these sets and give an interpretation of the results in terms of convex conical hulls of integer points. In particular, we apply the results to constructively describe growth rates of polynomials on any semialgebraic subset of the real plane. Moreover, we give the monomial generators for the algebra of bounded polynomials on basic semialgebraic set described by quasi-homogeneous inequalities. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Fix a semialgebraic set S in R n and a function g : S → R . We address the question on how to determine the power m such that for a polynomial f the function | f | / | g | m is bounded on S. As a consequence, we give explicit formulae for growth rate of a polynomial, i.e., the optimal power m when g = ‖ x ‖ , on a certain type of sets, called weighted tentacles, in terms of their support. To this aim, we study algebras of bounded polynomials for these sets and give an interpretation of the results in terms of convex conical hulls of integer points. In particular, we apply the results to constructively describe growth rates of polynomials on any semialgebraic subset of the real plane. Moreover, we give the monomial generators for the algebra of bounded polynomials on basic semialgebraic set described by quasi-homogeneous inequalities. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-025-00726-5
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      – Code: eng
        Text: English
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        PageCount: 23
        StartPage: 595
    Subjects:
      – SubjectFull: Semialgebraic sets
        Type: general
      – SubjectFull: Homogeneous polynomials
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Algebra
        Type: general
      – SubjectFull: Polyhedra
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      – TitleFull: Remarks on Growth Rates of Polynomials on Semialgebraic Sets: Remarks on Growth Rates of Polynomials on Semialgebraic Sets: M. Michalska.
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            – D: 01
              M: 04
              Text: Apr2025
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              Y: 2025
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            – TitleFull: Discrete & Computational Geometry
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