Budan Tables of Real Univariate Polynomials

André Galligo 1, 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : The Budan table of f collects the signs of the iterated derivative of f. We revisit the classical Budan-Fourier theorem for a univariate real polynomial f and establish a new connexity property of its Budan table. We use this property to characterize the virtual roots of f, (introduced by Gonzales-Vega, Lombardi, Mahe in 1998); they are continuous functions of the coecients of f. We also consider a property (P) of a polynomial f, which is generically satis ed, it eases the topological-combinatorial description and study of the Budan tables. A natural extension of the information collected by the virtual roots provides alternative representations of (P)-polynomials; while an attached tree structure allows a strati fication of the space of (P)-polynomials. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.
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André Galligo. Budan Tables of Real Univariate Polynomials. Journal of Symbolic Computation, Elsevier, 2013, 53, pp.64-80. ⟨10.1016/j.jsc.2012.11.004⟩. ⟨hal-00653756v2⟩

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