Multivariate Homogeneous Polynomial Research Articles

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in Rd can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.

Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

In this paper we associate to each multivariate polynomial $f$ that is homogeneous relative to a subset of its variables a series of polynomial families $P_\lambda (f)$ of $m$-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in $P_\lambda (f)$ is bounded from above by the circuit size of $f$. This provides a method for obtaining lower bounds for the circuit size of $f$ by proving $(s,r)$-(weak) elusiveness of the polynomial mapping associated with $P_\lambda (f)$. We discuss some algebraic methods for proving the $(s,r)$-(weak) elusiveness. We also improve estimates in the normal homogeneous-form of an arithmetic circuit obtained by Raz in \cite{Raz2009} which results in better lower bounds for circuit size (Lemma \ref{lem:cor38}, Remark \ref{rem:cor38}). Our methods yield non-trivial lower bound for the circuit size of several classes of multivariate homogeneous polynomials (Corollary \ref{cor:412}, Example \ref{ex:bi}).

Number of solutions of systems of homogeneous polynomial equations over finite fields

We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the homogeneous polynomials have the same degree that is not too large in comparison to the size of the finite field. We show that this conjecture holds in the affirmative if the number of polynomials does not exceed the total number of variables. This extends the results of Serre (1991) and Boguslavsky (1997) for the case of one and two polynomials, respectively. Moreover, it complements our recent result that the conjecture is false, in general, if the number of polynomials exceeds the total number of variables.

Resultant of an equivariant polynomial system with respect to the symmetric group

Given a system of n⩾2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols, it is proved that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

On a Conjecture of Tsfasman and an Inequality of Serre for the Number of Points of Hypersurfaces over Finite Fields

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality to an explicit formula for the maximum number of common solutions of a system of linearly independent multivariate homogeneous polynomials of the same degree with coefficients in a finite field. This conjecture is shown to be false, in general, but is also shown to hold in the affirmative in a special case. Applications to generalized Hamming weights of projective Reed-Muller codes are outlined and a comparison with an older conjecture of Lachaud and a recent result of Couvreur is given.

A product formula for multivariate Rogers–Szegő polynomials

Let \(H_n(t)\) denote the classical Rogers–Szegő polynomial, and let \(\tilde{H}_n(t_1, \ldots ,\) \( t_l)\) denote the homogeneous Rogers–Szegő polynomial in \(l\) variables, with indeterminate \(q\). There is a classical product formula for \(H_k(t)H_n(t)\) as a sum of Rogers–Szegő polynomials with coefficients being polynomials in \(q\). We generalize this to a product formula for the multivariate homogeneous polynomials \(\tilde{H}_n(t_1, \ldots , t_l)\). The coefficients given in the product formula are polynomials in \(q\) which are defined recursively, and we find closed formulas for several interesting cases. We then reinterpret the product formula in terms of symmetric function theory, where these coefficients become structure constants.

Linear subspaces of symmetric tensors whose non-zero elements have at least a prescribed rank

We introduce large vector spaces M of multivariate homogeneous polynomials with a prescribed lower bound for the rank of each non-zero element of M.

Sampling of homogeneous polynomials and approximating multivariate functions

Conditions for reconstruction of multivariate homogeneous polynomials from sets of sample values are introduced. In addition, it is shown that one can explicitly ob- tain the polynomial coefficients from the sample data by considering frames for spaces of symmetric tensors. Further, it is discussed how the reconstruction of homogeneous poly- nomials can be used to approximate certain smooth functions.

Spectra of symmetric tensors and m-root Finsler models

The present work studies spectral properties of multilinear forms attached to the Berwald-Moor, Chernov and Bogoslovsky locally Minkowski Finsler geometric structures of m-root type. We determine eigenvalues and the corresponding eigenvectors (of type Z, H and E) of these forms, in the framework of symmetric tensors and multivariate homogeneous polynomials. The geometric relevance of the spectral data is emphasized, and the existent relations between spectra, polyangles and Riesz-type associated 1-forms of the corresponding geometric models, are described. As well, the best rank-one approximation for the 4-dimensional Berwald-Moor and Chernov cases, is derived.

MIMO Instantaneous Blind Identification Based on Second-Order Temporal Structure and Steepest-Descent Method

This paper presents a new MIMO instantaneous blind identification algorithm based on second-order temporal property and steepest-descent method. Second-order temporal structure is reformulated in a particular way, such that each column of the unknown mixing matrix satisfies a system of nonlinear multivariate homogeneous polynomial equations. The nonlinear system is solved by an improved steepest-descent method. We convert the nonlinear system into an optimal problem by constructing an optimal objective function which includes a “valley-filled” item to obtain a valid solution. Our algorithm allows estimating the mixing matrix for scenarios with 4 sources and 3 sensors, etc. Finally, simulations show its effectiveness with more accurate solutions than the algorithm with homotopy method.

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.

MIMO Instantaneous Blind Identification Based on Second-Order Temporal Structure

Blind identification is a crucial subtask in signal processing problems such as blind signal separation (BSS) and direction-of-arrival (DOA) estimation. This paper presents a procedure for multiple-input multiple-output instantaneous blind identification based on second-order temporal properties of the signals, such as coloredness and nonstationarity. The procedure consists of two stages. First, based on assumptions on the second-order temporal structure (SOTS) of the source and noise signals, and using subspace techniques, the problem is reformulated in a particular way such that each column of the unknown mixing matrix satisfies a system of multivariate homogeneous polynomial equations. Then, this nonlinear system of equations is solved by means of a so-called homotopy method, which provides a general tool for solving (possibly nonexact) systems of nonlinear equations by smoothly deforming the known solutions of a simple start system into the desired solutions of the target system. Our blind identification procedure allows to estimate the mixing matrix for scenarios with more sources than sensors without resorting to sparsity assumptions, something that is often believed to be impossible when using only second-order statistics. In addition, since our algorithm does not require any assumption on the mixing matrix, also mixing matrices that are rank-deficient or even have identical columns can be identified. Finally, we give examples and performance results for speech source signals.

Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines

A real n -dimensional homogeneous polynomial f ( x ) of degree m and a real constant c define an algebraic hypersurface S whose points satisfy f ( x ) = c . The polynomial f can be represented by A x m where A is a real m th order n -dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial f , the hypersurface S and the tensor A . The rank is a nonnegative integer r less than or equal to n . When r is less than n , A is singular, f can be converted into a homogeneous polynomial with r variables by an orthogonal transformation, and S is a cylinder hypersurface whose base is r -dimensional. The eigenvalues of f , A and S always exist. The eigenvectors associated with the zero eigenvalue are either recession vectors or degeneracy vectors of positive degree, or their sums. When c ⁄ = 0 , the eigenvalues with the same sign as c and their eigenvectors correspond to the characterization points of S , while a degeneracy vector generates an asymptotic ray for the base of S or its conjugate hypersurface. The base index is a nonnegative integer d less than m . If d = k , then there are nonzero degeneracy vectors of degree k − 1 , but no nonzero degeneracy vectors of degree k . A linear combination of a degeneracy vector of degree k and a degeneracy vector of degree j is a degeneracy vector of degree k + j − m if k + j ≥ m . Based upon these properties, we classify such algebraic hypersurfaces in the nonsingular case into ten classes.

The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.

Properties of Multivariate Homogeneous Orthogonal Polynomials

It is well-known that the denominators of Padé approximants can be considered as orthogonal polynomials with respect to a linear functional. This is usually shown by defining Padé-type approximants from so-called generating polynomials and then improving the order of approximation by imposing orthogonality conditions on the generating polynomials. In the multivariate case, a similar construction is possible when dealing with the multivariate homogeneous Padé approximants introduced by the second author. Moreover it is shown here, that several well-known properties of the zeroes of classical univariate orthogonal polynomials, in the case of a definite linear functional, generalize to the multivariate homogeneous case. For the multivariate homogeneous orthogonal polynomials, the absence of common zeroes is translated to the absence of common factors.

Extrema of Multivariate Polynomials, Their Gradients and Directional Derivatives

O. D. Kellogg in [K28] established a connection between the supremum norms of a homogeneous polynomial and its gradient. We completed this result with a characterization of extrema in the bivariate case in [H94a] and announced the extension for multivariate homogeneous polynomials. The extension is presented in this paper. The generalization of the completed result to the case of arbitrary multivariate polynomials is also given here. The bivariate case of this contains, as special cases, the Bernstein and Markoff inequalities. Next, a well-known equality, involving suprema over directions of derivatives, is discussed. This relation turned out to be a dual to the above result in the homogeneous case (see [H94b]). On this basis, the sets of directions of suprema of the equality are characterized.

Massively parallel computations on many-variable polynomials

We show that a multivariate homogeneous polynomial can be represented on a hypercube in such a way that sums, products and partial derivatives can be performed by massively parallel computers. This representation is derived from the theoretical results of Beauzamy-Bombieri-Enflo-Montgomery [1]. The norm associated with it, denoted by [·], is itself a very efficient tool: when products of polynomials are performed, the best constant in inequalities of the form [PQ]≥C[P][Q] are provided, and the extremal pairs (that is, the pairs of polynomials for which the product is as small as possible) can be identified.

An algortihm for the inverse of a multivariate homogeneous polynomial of degree n

An algortihm for the inverse of a multivariate homogeneous polynomial of degree n

An algorithm for the inverse of a multivariate homogeneous polynomial of degreen

Various geodetic problems (the free nonlinear geodetic boundary value problem, the computation of Gaus-Kruger coordinates or UTM coordinates, the problem of nonlinear regression) demand theinversion of an univariate, bivariate, trivariate, in generalmultivariate homogeneous polynomial of degree n. The new algorithm which is oriented towardsSymbolic Computer Manipulation is based upon the algebraic power base computation with respect toKronecker-Zehfuβ product structure leading to the solution of a system oftriangular matrix equations: Only the first row of the inverse triangular matrix has to be computed. TheSymbolic Computer Manipulation program of the GKS algorithm is available from the authors.