Univariate Real Polynomials Research Articles

Approximate least common multiple of several polynomials using the ERES division algorithm

In this paper a numerical method for the computation of the approximate least common multiple (ALCM) of a set of several univariate real polynomials is presented. The most important characteristic of the proposed method is that it avoids root finding procedures and computations involving the greatest common divisor (GCD). Conversely, it is based on the algebraic construction of a special matrix which contains key data from the original set of polynomials and leads to the formulation of a linear system which provides the degree and the coefficients of the ALCM using low-rank approximation techniques and numerical optimization tools particularly in the presence of inaccurate data. The numerical stability and complexity of the method are analysed, and a comparison with other methods is provided.

Could René Descartes Have Known This?

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with the numbers of positive and negative roots. We provide a detailed information about possible non-realizable combinations up to degree 8 as well as a general conjecture about such combinations.

Counting multijoints

Let L1, L2, L3 be finite collections of L1, L2, L3, respectively, lines in R3, and J(L1,L2,L3) the set of multijoints formed by them, i.e. the set of points x∈R3, each of which lies in at least one line li∈Li, for all i=1,2,3, such that the directions of l1, l2 and l3 span R3. We prove here that |J(L1,L2,L3)|≲(L1L2L3)1/2 (which was previously known when L1, L2 and L3 are roughly the same),1 and we extend our results to multijoints formed by real algebraic curves in R3 of uniformly bounded degree, as well as by curves in R3 parametrised by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, as well as a discrete analogue of the endpoint multilinear Kakeya problem.

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes’ rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

Companion theorems to G. Szegö’s inequality for pairs of coefficients of bounded polynomials

We consider real univariate polynomials \(P_n\) of degree \( \le n \) from class $$\begin{aligned} \mathbf {C}_n = \{P_n:|P_n \left( \cos \displaystyle \frac{(n -i)\pi }{n}\right) |\le 1 \; \text{ for }\; 0\le i\le n \} \end{aligned}$$ which encompasses the unit ball of polynomials with respect to the uniform norm on \([- 1, 1]\). For pairs of consecutive coefficients of \( P_n(x) = \sum \nolimits _{k=0}^{n}a_kx^k\) there holds the inequality $$\begin{aligned} |a_{k-1}|+|a_k|\le |t_{n,k}|, \quad \text{ if }\; k\equiv n\; \text{ mod }\; 2, \end{aligned}$$ (1) where \(T_n(x)=\sum \nolimits _{k=0}^{n} t_{n,k}x^k\) is the \(n\)-th Chebyshev polynomial of the first kind. (1) implies Markov’s classical coefficient inequality of 1892 (Math. Ann. 77:213–258, 1916, p. 248) and goes back to Szego, but was made public by P. Erdos (Bull. Am. Math. Soc. 53:1169–1176, 1947, p. 1176) in 1947. We ask here: will the (nonzero) coefficients of \(T_n\) likewise majorize complementary pairs \(|a_k| + |a_{k+1}|\)? More generally: does there hold $$\begin{aligned} |a_k| + |a_j| \le |t_{n,k}| \quad \text{ for } \text{ all }\; P_n \in \mathbf {C_n}, \end{aligned}$$ (2) \(\text{ where }\; k K\), then (2) holds for all \(k\) with \(k_{*}\le k < j\), where the bound \(k_{*} < K\) is explicitly determined, and is optimal for \(n\le 43\). In Theorem 2.6 we return to G. Szego’s original inequality (1) and constructively prove the non - uniqueness of its extremizer \(\pm T_n\).

The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone

Abstract Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

Budan tables of real univariate polynomials

The Budan table of f collects the signs of the iterated derivatives of f. We revisit the classical Budan–Fourier theorem for a univariate real polynomial f and establish a new connectivity property of its Budan table. We use this property to characterize the virtual roots of f (introduced by Gonzalez-Vega, Lombardi, Mahé in 1998); they are continuous functions of the coefficients of f. We also consider a property (P) of a polynomial f, which is generically satisfied, it eases the topological-combinatorial description and study of 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 finite stratification of the space of polynomials with fixed degree.

Pólya-type polynomial inequalities in Orlicz spaces and best local approximation

Abstract We obtain an extension of Pólya-type inequalities for univariate real polynomials in Orlicz spaces. We also give an application to a best local approximation problem. MSC 2010: 41A10; 41A17.

On the degree and half-degree principle for symmetric polynomials

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte (2003) [15]. It says that a symmetric real polynomial F of degree d in n variables is positive on R n (or on R ≥ 0 n ) if and only if it is non-negative on the subset of points with at most max { ⌊ d / 2 ⌋ , 2 } distinct components. We deduce Timofte’s original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea that we are using to prove this statement is that of relating it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group S n this can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.

Computing the nearest polynomial with a zero in a given domain by using piecewise rational functions

For a real univariate polynomial f and a closed complex domain D whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial f ̃ such that f ̃ has a zero in D and ‖ f − f ̃ ‖ ∞ is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α ∈ C is a piecewise rational function of the real and imaginary parts of α . Thus, on C , the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C . Therefore, f ̃ can be constructed by using the property that f ̃ has a zero on C and computing the minimum distance on C . We analyze the asymptotic bit complexity of the method and show that it is of polynomial order in the size of the input.

Classifications of linear operators preserving elliptic, positive and non-negative polynomials

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer–Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [Borcea, Guterman, Shapiro, Preserving positive polynomials and beyond] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.

The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications

AbstractIn the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee‐Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity. © 2009 Wiley Periodicals, Inc.

Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in R N. In the book Fewnomials (6), A. Khovanskii gives bounds on the Betti numbers of varieties X in R n or in the positive orthant R n . These varieties include algebraic varieties, varieties defined by polynomial functions in the variables and exponentials of the variables, and also by even more general functions. Here, we consider only algebraic varieties. For these, Khovanskii bounds the sum b∗(X) of Betti numbers of X by a function that depends on n, the codimension of X, and the total number of monomials appearing (with non zero coefficients) in polynomial equations definingX. For real algebraic varieties X, the problem of bounding b∗(X) has a long history. O. A. Oleinik (8) and J. Milnor (7) used Morse theory to estimate the number of critical points of a suitable Morse function to obtain a bound. R. Thom (12) used Smith Theory to bound the mod-2 Betti numbers. This Smith-Thom bound has the form b∗(X) ≤ b∗(XC), where XC is the complex variety defined by the same polynomials as X. For instance, if X ⊂ R n is defined by polynomials of degree at most d, then Milnor's bound is b∗(X) ≤ d(2d − 1) n . If X ⊂ (R {0}) n is a smooth hypersurface defined by a polynomial with Newton polytope P , then the Smith-Thom bound is b∗(X) ≤ b∗(XC) = n! ¢ Vol(P ), where Vol(P ) is the usual volume of P. We refer to (10) for an informative history of the subject, and to the book (1) for more details. These bounds are given in term of degree or volume of a Newton polytope which are numerical deformation invariants of the complex variety XC. In contrast, the topology of a real algebraic variety depends on the coefficients of its defini ng equations, and in partic- ular, on the number of monomials involved in these equations. For instance, Descartes's rule of signs implies that the number of positive roots of a real univariate polynomial is less than its number of monomials, but the number of complex roots is equal to its degree. Khovanskii's bound can be seen as a generalization of this Descartes bound. It is smaller than the previous bounds when the defining equations have few monomials compared to their degrees. While it has always been clear that Khovanskii's bounds are unrealistically large, it appears very challenging to sharpen them. Some progress has been made recently for the number of non degenerate positive solutions to a system of n polynomials in n variables (3). Here, we bound b∗(X), when X is a fewnomial hypersurface. Suppose that X ⊂ R n is a smooth hypersurface defined by a Laurent polynomial with

Lee–Yang Problems and the Geometry of Multivariate Polynomials

We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Polya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory.

The nearest polynomial with a zero in a given domain

For a real univariate polynomial f and a closed domain D ⊂ C whose boundary C is represented by a piecewise rational function, we provide a rigorous method for finding a real univariate polynomial f ̃ such that f ̃ has a zero in D and ‖ f − f ̃ ‖ ∞ is minimal. First, we prove that if a nearest polynomial exists, there is a nearest polynomial f ̃ such that the absolute value of every coefficient of f − f ̃ is ‖ f − f ̃ ‖ ∞ with at most one exception. Using this property and the representation of C , we reduce the problem to solving systems of algebraic equations, each of which consists of two equations with two variables.

Pólya-Type Polynomial Inequalities in L P Spaces and Best Local Approximation

ABSTRACT In this paper, we obtain an extension of the Pólya inequality for univariate real polynomials in L p spaces and new estimates for certain class of measurable sets. Inequalities for complex polynomials are also considered. We give an application to a multipoint best local approximation problem for real and complex polynomials.

Testing Stability of 2-D Discrete Systems by a Set of Real 1-D Stability Tests

Stability of a two-dimensional (2-D) discrete system depends on whether a bivariate polynomial does not vanish in the closed exterior of the unit bi-circle. The paper shows a procedure that tests this 2-D stability condition by testing the stability of a finite collection of real univariate polynomials by a certain modified form of the author's one-dimensional (1-D) stability test. The new procedure is obtained by telepolation (interpolation) of a 2-D tabular test whose derivation was confined to using a real form of the underlying 1-D stability test. Consequently, unlike previous telepolation-based tests, the procedure requires the testing of real instead of complex univariate polynomials. The proposed test is the least-cost procedure to test 2-D stability with real polynomial 1-D stability tests and real arithmetic only.

Durand-Kerner method for the real roots

Given a univariate real polynomial, this paper considers calculating all the real zero-points of the polynomial simultaneously. Among several numerical methods for calculating zero-points of a univariate polynomial, Durand-Kerner method is quite useful because it is most stable to converge to the zero-points. On the basis of Durand-Kerner method, we propose two methods for calculating the real zero-points of a univariate polynomial simultaneously. Convergence and error analysis of our methods are discussed. We compared our methods, the original Durand-Kerner method and Newton’s method and found that 1) our methods are more stable than Newton’s method but less than the original Durand-Kerner method, and 2) they are more efficient than the original Durand-Kerner method but less than Newton’s method. We conclude that our methods are useful when good initial values of the zero-points are known.

On the Geometry of Graeffe Iteration

A new version of Graeffe's algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is called the renormalized Graeffe iteration. It is globally convergent, with probability 1. All quantities involved in the computation are bounded once the initial polynomial is given (with probability 1). This implies remarkable stability properties for the new algorithm, thus overcoming known limitations of the classical Graeffe algorithm. If we start with a degree-d polynomial, each renormalized Graeffe iteration costs O(d2) arithmetic operations, with memory O(d). A probabilistic global complexity bound is given. The case of univariate real polynomials is briefly discussed. A numerical implementation of the algorithm presented herein allows us to solve random polynomials of degree up to 1000.

Virtual roots of real polynomials

The fact that a real univariate polynomial misses some real roots is usually overcome by considering complex roots, but the price to pay for, is a complete loss 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.