Bounds for Absolute Positiveness of Multivariate Polynomials

A multivariate polynomialP(x1, �,xn) with real coefficients is said to beabsolutely positivefrom a real numberBiff it and all of its non-zero partial derivatives of every order are positive forx1,�,xn�B. We call suchBaboundfor the absolute positiveness ofP. This paper provides a simple formula for computing such bounds. We also prove that the resulting bounds are guaranteed to be close to the optimal ones.

[1]  Matsusaburô Fujtwara.  Über die Wurzeln der algebraischen Gleichungen , 1915 .

[2]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[3]  Wolfgang Küchlin,et al.  Parallel Real Root Isolation Using the Coefficient Sign Variation Method , 1990, CAP.

[4]  D. Grigor'ev.  Complexity of deciding Tarski algebra , 1988 .

[5]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[6]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[7]  P. Borwein,et al.  Polynomials and Polynomial Inequalities , 1995 .

[8]  Dima Grigoriev,et al.  Complexity of Deciding Tarski Algebra , 1988, J. Symb. Comput..

[9]  George E. Collins,et al.  An efficient algorithm for infallible polynomial complex root isolation , 1992, ISSAC '92.

[10]  James Renegar.  On the computational complexity and geome-try of the first-order theory of the reals , 1992 .

[11]  N. A C H U M D E R S H O W I T Z.  Termination of Rewriting' , 2022 .

[12]  George D. Birkhoff,et al.  An elementary double inequality for the roots of an algebraic equation having greatest absolute value , 1915 .

[13]  Lee E. Heindel.  Algorithms for exact polynomial root calculation , 1970, SIGS.

[14]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[15]  Rüdiger Loos,et al.  Polynomial real root isolation by differentiation , 1976, SYMSAC '76.

[16]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[17]  A. Cohn,et al.  Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise , 1922 .

[18]  Pierre Lescanne,et al.  Termination of Rewriting Systems by Polynomial Interpretations and Its Implementation , 1987, Sci. Comput. Program..

[19]  Hoon Hong,et al.  Improvements in cad-based quantifier elimination , 1990 .

[20]  George E. Collins,et al.  A hybrid method for high precision calculation of polynomial real roots , 1993, ISSAC '93.

[21]  Dennis Soule Arnon.  Algorithms for the geometry of semi-algebraic sets , 1981 .

[22]  J. Urgen Giesl.  Generating Polynomial Orderings for Termination Proofs ? , 1995 .

[23]  Scott McCallum,et al.  An Improved Projection Operation for Cylindrical Algebraic Decomposition , 1985, European Conference on Computer Algebra.

[24]  Jeremy R. Johnson,et al.  Real algebraic number computation using interval arithmetic , 1992, ISSAC '92.

[25]  Joachim Steinbach.  Proving Polynomials Positive , 1992, FSTTCS.

[26]  W. Rogosinski,et al.  The Geometry of the Zeros of a Polynomial in a Complex Variable , 1950, The Mathematical Gazette.

[27]  Gradimir V. Milovanovic,et al.  Topics in polynomials - extremal problems, inequalities, zeros , 1994 .

[28]  Jeremy Johnson,et al.  Algorithms for polynomial real root isolation , 1992 .

[29]  R. D. Carmichael,et al.  Note on the roots of algebraic equations , 1914 .

[30]  Alkiviadis G. Akritas,et al.  Polynomial real root isolation using Descarte's rule of signs , 1976, SYMSAC '76.

[31]  Scott McCallum.  An improved projection operation for cylindrical algebraic decomposition (computer algebra, geometry, algorithms) , 1984 .

[32]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[33]  Joos Heintz,et al.  On the Complexity of Semialgebraic Sets , 1989, IFIP Congress.