Acta Mathematica Vietnamica

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Algebra of Polynomials Bounded on a Semi-algebraic Set $[f \leq r]$

$icon-email$ Du Thu Trang , Ho Minh Toan , Nguyen Thi Hong

Abstract

The algebra of polynomials in $R [x]$ which are bounded on a semi-algebraic set determined by a polynomial inequality $f (x) \leq r$ with $f (0) = 0$ is studied and the case when it is generated by a finite set of monomials is discussed. A large class of polynomials which are asymptotic to finitely many monomials (including nondegenerate polynomials) is introduced and the algebra of polynomials bounded on $[f \leq r]$ can be determined by a cone and is independent on $r > 0,$ where $f$ belongs to this class. Note that the set of all polynomials whose supports lie in a given closed convex cone in the first quadrant forms an algebra generated by a finite set of monomials. In other cases, we can give upper and lower bounds of the algebra via outer normal cones of the faces of the Newton polyhedron. As a consequence, some sufficient conditions which ensure that the algebra under consideration is generated by finitely many monomials is given.

Acta Mathematica Vietnamica

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Algebra of Polynomials Bounded on a Semi-algebraic Set [f≤r]

Abstract

Algebra of Polynomials Bounded on a Semi-algebraic Set $[f \leq r]$