Some Inequalities for Polynomials With a Prescribed Zero

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.

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.

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.

We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent (LE) among all trajectories in a specified set. A minimisation problem is formulated whose infimum is equal to the maximal LE, provided that trajectories of interest remain in a compact set. The minimisation is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case—i.e. when the ODE’s right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials—the minimisation can be relaxed into a computationally tractable polynomial optimisation problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimisation problems of increasing computational cost whose infima converge from above to the maximal LE, at least when the set of interest is compact. For illustration, we carry out such polynomial optimisation computations for two chaotic examples: the Lorenz system and the Hénon–Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.

Let f ( z ) f(z) be a polynomial of degree n. Given that f ( z ) f(z) has a zero on the circle | z | = ρ ( 0 > ρ > ∞ ) |z| = \rho \;(0 > \rho > \infty ) , we estimate | f ( 0 ) | |f(0)| and \[ ( ( 2 π ) − 1 ∫ 0 2 π | f ( e i θ ) | 2 d θ ) 1 / 2 {({(2\pi )^{ - 1}}\int _0^{2\pi }|f({e^{i\theta }}){|^2}d\theta )^{1/2}} \] in terms of max | z | = 1 | f ( z ) | {\max _{|z| = 1}}|f(z)| . We also consider some other related problems.

Primitive ideals and orbital integrals in complex exceptional groups

Let P(z) be a polynomial of degree n 1 . In this paper we consider a more general problem of investigating the dependence of maximum of ∣∣∣P(Rz) − αP(z) + β {( R + 1 2 )n − |α| } P(z) ∣∣∣, R 1, on the maximum of |P(z)| on |z| = 1 where α , β are arbitrary complex numbers with |α| 1 , |β | 1 and obtain certain sharp compact generalizations of well-known Bernsteintype polynomial inequalities. Mathematics subject classification (2000): 30C10, 30A10.

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P 0 (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

Let be a polynomial of degree n and let p′(z) be the derivative of p(z). Then it is well known that and .In this paper we obtain inequalities analogous to these inequalities for polynomials satisfying and with all their coefficients lying in a sector. Our results generalize and sharpen some of the known results.

The complexity of quantum query algorithms computing Boolean functions is strongly related to the degree of the algebraic polynomial representing this Boolean function. There are two related difficult open problems. First, Boolean functions are sought for which the complexity of exact quantum query algorithms is essentially less than the complexity of deterministic query algorithms for the same function. Second, Boolean functions are sought for which the degree of the representing polynomial is essentially less than the complexity of deterministic query algorithms. We present in this paper new techniques to solve the second problem.

We study the problem of estimating an affine transformation with prescribed determinant from point pairs in dimension . The determinant constraint is nonconvex and challenging to handle. We propose a method to reformulate the problem into a system of quadratics and one polynomial of degree in unknowns. We propose a solution method for this system and a procedure to retrieve the single or multiple affine transformation solutions. We show that the related problem of finding the closest ‐simplex of prescribed volume and orientation to a given arbitrary ‐simplex is solved as a special case of the proposed method. We report experimental results demonstrating the use of the proposed transformations in three applications.

On the density of polyharmonic splines

Let $$\mathcal{P}_n^c$$ denote the set of all algebraic polynomials of degree at most n with complex coefficients. Let $$\begin{aligned} D^+ := \{z \in \mathbb {C}: |z| \le 1, \text { Im}(z) \ge 0\}\,. \end{aligned}$$For integers $$0 \le k \le n$$ let $$\mathcal{F}_{n,k}^c$$ be the set of all polynomials $$P \in \mathcal{P}_n^c$$ having at least $$n-k$$ zeros in $$D^+$$. Let $$\begin{aligned} \Vert f\Vert _A := \sup _{z \in A}{|f(z)|} \end{aligned}$$for complex-valued functions defined on $$A \subset \mathbb {C}$$. We prove that there are absolute constants $$c_1 > 0$$ and $$c_2 > 0$$ such that $$\begin{aligned} c_1 \left( \frac{n}{k+1}\right) ^{1/2} \le \inf _{P}{\frac{\Vert P^{\prime }\Vert _{[-1,1]}}{\Vert P\Vert _{[-1,1]}}} \le c_2 \left( \frac{n}{k+1}\right) ^{1/2} \end{aligned}$$for all integers $$0 \le k \le n$$, where the infimum is taken for all $$0 \not \equiv P \in \mathcal{F}_{n,k}^c$$ having at least one zero in $$[-1,1]$$. This is an essentially sharp reverse Markov-type inequality for the classes $$\mathcal{F}_{n,k}^c$$ extending earlier results of Turan and Komarov from the case $$k=0$$ to the cases $$0 \le k \le n$$.

For the polynomial P(z) = n \sum j=0 cjzj of degree n having all its zeros in | z| \leq k, k \geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P \prime (z)| \geq n \biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) \biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n \geq 2.

In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of a random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due to Kim and Vu. The main advantage of our method in comparison with the existing ones is a wide range of random variables we can handle and bounds for previously intractable regimes of high degree polynomials and small expectations. On the negative side we show that even for boolean random variables each term in our concentration inequality is tight.