Complex rectangular transforms

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.

A new approach to the symbolic factorization of multivariate polynomials

In the recent papers, a new efficient probabilistic semi-numerical absolute (i.e., complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It is based on a simple property of the monomials arising after a generic linear change of coordinates for bivariate polynomials and on a deep result of complex algebraic geometry. Here, we consider the a priori simpler problem of factorization over the field of reals. We briefly review our algorithm for complex factorization and adapt it to solving the problem on the field of reals. This allows us to spare a significant part of the computations and to improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of \(\mathbb{Q}\). Bibliography: 15 titles.

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.The well-known Kolmogorov theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition adjoins the completeness problem (see the formulation of this theorem below).The purpose of this paper is the following problem: is there an analogue of Kolmogorov’s theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition for functional systems of polynomial functions and functional systems of rational functions?It turns out that the problem posed (an analogue of Kolmogorov’s theorem) has a negative answer for functional systems of polynomial functions with natural and integer coefficients, and for functional systems of polynomial functions with rational and real coefficients and for functional systems of rational functions with rational and real coefficients - the answer is positive. These theorems are the main results of this paper.

Maple 2019 has a new multivariate polynomial factorization algorithm for factoring polynomials in \(\mathbb {Z}[x_1,x_2,...,x_n]\), that is, polynomials in n variables with integer coefficients. The new algorithm, which we call MTSHL, was developed by the authors at Simon Fraser University. The algorithm and its sub-algorithms have been published in a sequence of papers [3, 4, 5]. It was integrated into the Maple library in early 2018 by Baris Tuncer under a MITACS internship with Maplesoft. MTSHL is now the default factoring algorithm in Maple 2019.

For odd square-free $n > 1$ the cyclotomic polynomial ${\Phi _n}(x)$ satisfies the identity of Gauss, \[ 4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2.\] A similar identity of Aurifeuille, Le Lasseur, and Lucas is \[ {\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2\] or, in the case that n is even and square-free, \[ \pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2.\] Here, ${A_n}(x), \ldots ,{D_n}(x)$ are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require $O({n^2})$ arithmetic operations and work over the integers. We also give explicit formulae and generating functions for ${A_n}(x), \ldots ,{D_n}(x)$, and illustrate the application to integer factorization with some numerical examples.

For odd square-free n > 1 n > 1 the cyclotomic polynomial Φ n ( x ) {\Phi _n}(x) satisfies the identity of Gauss, \[ 4 Φ n ( x ) = A n 2 − ( − 1 ) ( n − 1 ) / 2 n B n 2 . 4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. \] A similar identity of Aurifeuille, Le Lasseur, and Lucas is \[ Φ n ( ( − 1 ) ( n − 1 ) / 2 x ) = C n 2 − n x D n 2 {\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 \] or, in the case that n is even and square-free, \[ ± Φ n / 2 ( − x 2 ) = C n 2 − n x D n 2 . \pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. \] Here, A n ( x ) , … , D n ( x ) {A_n}(x), \ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O ( n 2 ) O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n ( x ) , … , D n ( x ) {A_n}(x), \ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.

A central tool in the study of systems of linear equations with integer coefficients is the generalised von Neumann theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms ∥f∥Us+1[N] of the weight f. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the Mobius function over certain sequences. The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues.

Let F_1,ldots ,F_R be quadratic forms with integer coefficients in n variables. When nge 9R and the variety V(F_1,ldots ,F_R) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V(F_1,ldots ,F_R). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n> d2^dR. In the case that dge 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.

This paper describes a method for calculating the roots of an equation with real coefficients. While the basic approach is not new, two relationships are presented which are not widely known : (1) a guaranteed initial approximation of a root ; (2) a method of making this initial approximation the smallest root of the equation under study. A method of determining the magnitude of an axis shift is also presented.

The problem of factorizing a multivariable or multidimensional (m-D) polynomialf (z 1,z 2, ...,z m ), with real or complex coefficients and independent variables, into a number of m-D polynomial factors that may involve any independent variable or combination of them is considered. The only restriction imposed is that all factors should be linear in one and the same variable (sayz 1). This type of factorization is very near to the most general type: $$\begin{gathered} f(z_1 ,z_2 , \ldots ,z_m ) = \prod\limits_{i = 1}^{N_1 } {\lbrack {\mathop \sum \limits_{\epsilon _1 = 0}^{\epsilon _{i,1} } \cdots \mathop \sum \limits_{\epsilon _m = 0}^{\epsilon _{i,m} } a_{i;\epsilon _1 ,\epsilon _2 , \ldots ,\epsilon _m } z_1^{\epsilon _1 } \cdots z_m^{\epsilon _m } + c_i } \rbrack} \hfill \\ (\epsilon _1 , \ldots ,\epsilon _m ) \neq (0, \ldots ,0) \hfill \\ \end{gathered}$$ and appears to be the most general type available. The method is first briefly sketched for the convenience of the reader, and then is presented in detailed form through a number of theorems. These theorems provide a clear algorithmic way for the factorization, which may be automated via a suitable computer code. The factorization of m-D polynomials simplifies the stability analysis and the realization of m-D systems, as well as the solution of distributed parameters systems.

Let [special characters omitted] be function of [special characters omitted], 0 [special characters omitted], and [special characters omitted], that takes m x m nonnegative matrix values and such that [special characters omitted] for all [special characters omitted]) and all [special characters omitted]. Denote by [special characters omitted] the space of N-vector valued functions continuous on the closed unit disc in [special characters omitted] and analytic inside the disc whose Taylor series expansion about 0 has real coefficients. We consider the following problem: [special characters omitted]where [special characters omitted] is a subset of [special characters omitted] that may depend on given continuous N-vector valued functions [special characters omitted] whose entries have real Fourier coefficients, on given continuous scalar valued functions [special characters omitted] with real Fourier coefficients, and on given real constants cl. The set [special characters omitted] we consider takes one of the following forms:[special characters omitted]We establish optimality conditions for each of these three cases that must be satisfied by a solution of the (CONSTRAINED-OPT) problem. These optimality conditions can be used for finding good candidates to solutions of the (CONSTRAINED-OPT), and we discuss algorithms for finding such candidates for [special characters omitted] as in (1) and (2). Numerical examples are presented. We also consider the problem of finding a spectral factorization of a given, possibly low rank, positive semidefinite matrix valued function on the unit circle of the complex plane. We derive operator equations that must be satisfied by solutions to the spectral factorization problem for either the low or the full rank cases. The equations we derive can be solved numerically using Newton's method. We prove that sufficient conditions for local quadratic convergence of the Newton's method are satisfied. Numerical examples are given. Finally, we develop an implementation of a high level algorithm of Peller-Young for finding superoptimal approximants to a given matrix valued continuous function. Our implementation uses spectral factorization as well as other tools that we developed for this purpose. Also a numerical example is presented.

A new package for factoring polynomials with integer coefficients is described which yields significant improvements over previous implementations in both time and space requirements. For multivariate problems, the package features an inexpensive method for early detection and correction of spurious factors. This essentially solves the multivariate extraneous factor problem and eliminates the need to factor more than one univariate image, except in rare cases. Also included is an improved technique for coefficient prediction which is successful more frequently than prior versions at short-circuiting the expensive multivariate Hensel lifting stage. In addition some new approaches are discussed for the univariate case as well as for the problem of finding good integer substitution values. The package has been implemented both in Scratchpad II and in an experimental version of muMATH.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree d in n variables over the field of q elements, one desires an algorithm whose running time is a polynomial function of dn log(q). (Here we assume d ⩾ 2, for otherwise the problem is easy.) The case n = 1 is related to univariate polynomial factorisation and is comparatively straightforward. When n = 2 one is counting points on curves, and the method of Schoof and Pila yields a complexity of log ⁡ ( q ) C d , where the function Cd depends exponentially on d. For arbitrary n, the theorem of the author and Wan gives a complexity which is a polynomial function of (pdn log(q))n, where p is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of pdn log(q). In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form Zp − Z = f, where the polynomial f has a diagonal leading form. The method utilises a relative p-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let f be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime p, outputs the number of solutions to the congruence equation f = 0 op, and runs in {\mathcal O}(p^{2 + \varepsilon}) bit operations, for any ε > 0. This improves upon the elementary estimate of {\mathcal O}(p^{n-1 + \varepsilon}) bit operations, where n is the number of variables, which can be achieved using Berlekamp's root counting algorithm. 2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.

The issues related to the problem of minimizing hardware costs in the digital algorithms’ hardware and software implementation for the discrete signals’ frequency selection on programmable logic devices (PLD) and special processor microelectronic devices are considered. Possible ways to solve these problems are described based on the computational algorithms for non-recursive digital filtering (NDF) and difference digital filtering with integer coefficients use. A necessary and sufficient condition is given for using a computational algorithm’s hardware and software implementation of discrete signals’ difference digital filtering for their multi-stage discrete Fourier transform without performing arithmetic multiplication operations.