Virtual roots of real polynomials
Copyright policy )Virtual roots of real polynomials
The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing roots which keep sign properties and which extend of course the existing roots. Moreover these "virtual roots" are the values of semialgebraic continuous -- rather uniformly -- functions defined on the set of monic polynomials. We present some applications.
tiny corrections to the original paper
- French National Centre for Scientific Research France
- University of Franche-Comté France
- University of Rennes France
- Laboratoire de Mathématiques France
- University of Rennes 1 France
virtual root, Algebra and Number Theory, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Polynomials in real and complex fields: factorization, 14P05 (14Q05), Pierce-Birkhoff conjecture, spline, 004, [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, Spline approximation, Effectivity, complexity and computational aspects of algebraic geometry, semi-algebraic set, FOS: Mathematics, real root, real polynomial, Semialgebraic sets and related spaces, Algebraic Geometry (math.AG)
virtual root, Algebra and Number Theory, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Polynomials in real and complex fields: factorization, 14P05 (14Q05), Pierce-Birkhoff conjecture, spline, 004, [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, Spline approximation, Effectivity, complexity and computational aspects of algebraic geometry, semi-algebraic set, FOS: Mathematics, real root, real polynomial, Semialgebraic sets and related spaces, Algebraic Geometry (math.AG)
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