相关论文: On Descartes' rule of signs for hyperbolic polynom…

We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where…

A real univariate polynomial with all roots real is called hyperbolic. By Descartes' rule of signs for hyperbolic polynomials (HPs) with all coefficients nonvanishing, a HP with $c$ sign changes and $p$ sign preservations in the sequence of…

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…

A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with…

We consider real univariate polynomials with all roots real. Such a polynomial with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has $c$ positive and $p$ negative roots counted with multiplicity. Suppose…

We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive…

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule…

The classical Descartes' rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers…

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the…

The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration…

By Descartes' rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $pos\leq c$ positive and $neg\leq p$…

For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers $pos$ of positive and $neg$ of negative roots (counted with multiplicity) are majorized…

We present an optimal version of Descartes' rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with n+2 monomials. This sharp upper bound is given in terms of the sign…

Given a real univariate degree $d$ polynomial $P$, the numbers $pos_k$ and $neg_k$ of positive and negative roots of $P^{(k)}$, $k=0$, $\ldots$, $d-1$, must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's…

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface…

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and…

We develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's "polygon rule".

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…