Counting zeros of generalized polynomials: Descartes’ rule of signs and Laguerre’s extensions
نویسنده
چکیده
A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as “Descartes’ rule of signs”. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie, which appeared in 1637. It can be proved by a method based on factorization, but, again, just as easily by deduction from Rolle’s theorem.
منابع مشابه
A Problem of P olya Concerning Polynomials Which Obey Descartes ' Rule of Signs
We formulate and duscuss two open problems. The rst one is due to PP olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karlin and states that certain Hankel determinants associated with an entire function in the Laguerre-...
متن کاملOn zero curves of bivariate polynomials
With the emergence of algebraic curves and surfaces in geometric modelling [2,4,6,11,12, 21,22] it is important to be able to predict how many connected components the zero set of a multivariate function has in terms of its coefficients. It would be especially useful to find a condition which ensures that the zero set is a single curve or surface. For univariate polynomials Descartes’ Rule of S...
متن کاملKarlin’s Conjecture and a Question of Pólya
The paper answers an old question of Pólya involving Descartes’ Rule of Signs and a related conjecture of Karlin involving the signs of Wronskians of entire functions and their derivatives. Counterexamples are given along with classes of functions for which the conjecture is valid. 0. Introduction. The purpose of this paper is to answer an old unsolved question of Pólya (c. 1934) and to resolve...
متن کاملPolynomials with Nonnegative Coefficients
The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independ...
متن کاملDescartes' Rule of Signs for Radial Basis Function Neural Networks
We establish versions of Descartes' rule of signs for radial basis function (RBF) neural networks. The RBF rules of signs provide tight bounds for the number of zeros of univariate networks with certain parameter restrictions. Moreover, they can be used to infer that the Vapnik-Chervonenkis (VC) dimension and pseudodimension of these networks are no more than linear. This contrasts with previou...
متن کامل