Markov–Bernstein Type Inequalities for Multivariate Polynomials on Sets with Cusps

In this work, we discuss generalizations of the classical Bernstein and\nMarkov type inequalities for polynomials and we present some new inequalities\nfor the $k$th Fr\\'echet derivative of homogeneous polynomials on real and\ncomplex $L_{p}(\\mu)$ spaces. We also give applications to homogeneous\npolynomials and symmetric multilinear mappings in $L_{p}(\\mu)$ spaces. Finally,\nBernstein's inequality for homogeneous polynomials on both real and complex\nHilbert spaces has been discussed.\n

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

Let P(z) be a polynomial of degree n and for a complex number , let DP(z) = nP(z) + (alpha − z) P′(z) denote the polar derivative of the polynomial P(z) with respect to alpha . In this paper we establish some Lr inequalities for the polar derivative of a polynomialwith restricted zeros. Our results not only generalizes some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.

Polynomial Inequalities on Measurable Sets and Their Applications II. Weighted Measures

Semialgebraic computations, i.e., the manipulation of sets and logical conditions defined by polynomial inequalities in real variables, are an essential ”primitive” in the analysis and design of hybrid dynamical systems. Fundamental tasks such as reachability analysis, abstraction verification, and the computation of stability and performance certificates, all use these operations extensively, and can quickly become the computational bottleneck in the design process. Although there is a well-developed body of both basic theory and algorithms for these tasks, the practical performance of most available methods is still far from being satisfactory on real-world problems. While there are several possible causes for this (besides the NP-hardness of the task), a sensible explanation lies in the purely algebraic nature of the usual methods, as well as the insistence on exact (as opposed to approximate or ”relaxed”) solutions. For these reasons, there is a strong interest in the development of efficient techniques for (perhaps restricted) classes of semialgebraic problems. In this talk we review the basic elements and present several new results on the SOS approach to semialgebraic computations, that combines symbolic and numerical techniques from real algebra and convex optimization. Its main defining feature is the use and computation of sum of squares (SOS) decompositions for multivariate polynomials via semidefinite programming. These are extended, using the Positivstellensatz, to structured infeasibility certificates for polynomial equations and inequalities. The developed techniques unify and generalize many well-known existing methods. In particular, we will discuss semialgebraic problems with at most two quantifier alternations (i.e., classical polynomial optimization problems as well as the related games and minimax problems). As an example, we will solve in detail a class of zero-sum two-person games with an infinite number of pure strategies, where the payoff function is a polynomial expression of the actions of the players. Although particular emphasis will be given to the hybrid systems viewpoint, the basic ideas and algorithms, as well as these recent extensions, will be illustrated with examples drawn from a broad range of related domains, including dynamical systems and geometric theorem proving.

The classical Bernstein pointwise estimate of the (first) derivative of a univariate algebraic polynomial on an interval has natural extensions to the multivariate setting. However, in several variables the domain of boundedness, even if convex, has a considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled the case of a centrally symmetric convex body by a method we may call "the method of inscribed ellipses." On the other hand, for the general case of nonsymmetric convex bodies we are only within a constant factor of an exact inequality. The best known results suggest relevance of the generalized Minkowski functional, and a natural conjecture for the exact Bernstein factor was formulated with this geometric quantity. This work deals with the most natural and simple nonsymmetric case, that of a standard simplex in, and computes the exact yield of the method of inscribed ellipses. Although the known general estimates of the Bernstein factor are improved for the simplex here, we find that not even the exact yield of the inscribed ellipse method reaches the conjecture. However, we also show that for an arbitrary convex body the subset of ridge polynomials satisfies the conjecture.

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.

There are fine extensions of the univariate Bernstein-Szegő inequality for multivariate polynomials considered on a convex domain K. The current one estimates the gradient of the polynomial P at a point x ∈ K by constant times degree, ‖P‖C(K) and a geometrical factor. The best constant is within [2, 2√2]. In this note we disprove the conjecture (based on some particular cases) that the best constant is 2.

In this paper, certain new results concerning the maximum modulus of the polar derivative and the generalized polar derivative of a polynomial with restricted zeros are obtained. These estimates strengthen some well known inequalities for polynomial due to Tur n, Rather and Dar, Rather, Ali, Shafi and Dar and others.

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. Bibiography: 13 titles.

We link the study of positive quantum maps, block positive operators and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.

In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space $${{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}$$ with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials.

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.