On Multivariate Descartes' Rule - A Counterexample

Abstract: The purpose of this short note is to give a counterexample to a widely known conjecture on a multivariate version of Descartes' rule [D] proposed by I. Itenberg and M.-F. Roy [Roy]. The famous Descartes' rule states that the number of positive real zeroes of a univariate polynomial is not greater than the number of sign changes in the list of its coefficients. \item{[D]} Descartes, R.: Geometrie, 1636. In: A source book in Mathematics, Massachusetts, Harvard University Press, 1969, 90-93. \item{[R]} Itenberg, I.; Roy, M.-F.: Multivariate Descartes' rule. Beiträge zur Algebra und Geometrie 37 (1996), 337-346.

Keywords: multivariate version; Descartes rule; zeroes of polynomials

Classification (MSC91): 65H10,50F14,CR: G1.5.

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