Univariate Polynomials Research Articles

A functional equation with polynomial solutions and application to Neural Networks

We construct and discuss a functional equation with contraction property. The solutions are real univariate polynomials. The series solving the natural fixed point iterations have immediate interpretation in terms of Neural Networks with recursive properties and controlled accuracy.

Inequalities for the maximum modulus of univariate constrained polynomials

This paper deals with the problem of finding some upper bound estimates for the maximum modulus of a univariate complex polynomial on a disk under certain constraints on the zeros and on the functions involved. A variety of interesting results follow as special cases from our general results.

Structured backward errors in linearizations

A standard approach to calculate the roots of a univariate polynomial is to compute the eigenvalues of an associated confederate matrix instead, such as, for instance, the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure-exploiting QR algorithm. In [J. L. Aurentz et al., Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942–973] it was shown that a structure-exploiting QR algorithm for companion matrices leads to a structured backward error in the companion matrix. The proof relied on decomposing the error into two parts: a part related to the recurrence coefficients of the basis (a monomial basis in that case) and a part linked to the coefficients of the original polynomial. In this article we prove that the analysis can be extended to other classes of comrade matrices. We first provide an alternative backward stability proof in the monomial basis using structured QR algorithms; our new point of view shows more explicitly how a structured, decoupled error in the confederate matrix gets mapped to the associated polynomial coefficients. This insight reveals which properties have to be preserved by a structure-exploiting QR algorithm to end up with a backward stable algorithm. We will show that the previously formulated companion analysis fits into this framework, and we analyze in more detail Jacobi polynomials (comrade matrices) and Chebyshev polynomials (colleague matrices).

When is Freshman's Dream Actually True?

We address the problem of determining what points in a field satisfy Freshman's Dream, or equivalently, when a monomial behaves additively. It is conjectured that the only additive points over the rational numbers are trivial. In the case of finite fields, we generalize well-known results about uni-variate polynomials to bivariate homogeneous polynomials in order to count the number of additive points.

Multiparameter Eigenvalue Problems and Shift-invariance

We discuss four eigenvalue problems of increasing generality and complexity: rooting a univariate polynomial, solving the polynomial eigenvalue problem, rooting a set of multivariate polynomials and solving multi-parameter eigenvalue problems. In doing so, we provide a unifying framework for solving these eigenvalue problems, where we exploit properties of (block-) (multi-) shift-invariant subspaces and use multi-dimensional realization algorithms.

Systems of Polynomial Equations, Higher-order Tensor Decompositions, and Multidimensional Harmonic Retrieval: A Unifying Framework. Part I: The Canonical Polyadic Decomposition

We propose a multilinear algebra framework to solve systems of polynomial equations with simple roots. We translate connections between univariate polynomial root-finding, eigenvalue decompositions...

Visual Performance of a Polynomial Extended Depth of Focus Intraocular Lens

Purpose: To clinically evaluate a new extended depth of focus intraocular lens (ISOPURE, PhysIOL) with optic design modification based on a unique polynomial concept to improve intermediate vision while keeping the quality of distance vision equal to a monofocal lens. Methods: 18 patients (11 female, 7 male, mean age of 69.4 years) with bilateral cataract and regular corneal astigmatism ≤ 1.0 D underwent bilateral cataract surgery with ISOPURE implantation. Patients were followed for up to 6 months. Measured parameters were uncorrected (UDVA) and corrected distance visual acuity (CDVA), uncorrected (UIVA) and distance-corrected intermediate visual acuity at 80 cm and 66 cm (DCI80VA, DCI66VA) subjective refraction, defocus curve, tolerance of cylinder induction, and contrast sensitivity. The data from all implanted eyes (all-eyes) and a subset only including the first eye implanted for each patient were analysed. Results: The mean manifest refraction spherical equivalent (MRSE) decreased from 1.05 D pre-operatively to ?0.15 D at the 4 - 6 month assessment, with 80.6% of eyes within ±0.50 D of emmetropia. At the final follow-up, mean (SD) monocular CDVA was ?0.06 (0.04) logMAR, DCI80VA was 0.18 (0.08) logMAR and DCI66VA was 0.27 (0.13) logMAR. Despite a cylinder induction of ?0.50 D, uncorrected distance visual acuity of 0.02 logMAR was still achieved. Conclusion: The ISOPURE intraocular lens provides excellent distance corrected visual acuity for far and intermediate distances along with high contrast sensitivity and good tolerance of residual refractive cylinder.

Cycle Neighbor Polynomial of Graphs

In this paper, a new univariate graph polynomial called Cycle Neighbor Polynomial \(CN[G;x]\) of a graph \(G\) is introduced. We obtain some interesting properties of this polynomial and compute cycle neighbor polynomial of some specific graphs.

Computing Igusa’s local zeta function of univariates in deterministic polynomial-time

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression for the number of roots $N_{k}(f)$. Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly($|f|, \log p$) time algorithm to compute $Z_{f,p}(s)$. Previously, an algorithm was known only in the case when $f$ completely splits over $\mathbb{Q}_p$; it required the rational roots to use the concept of generating function of a tree (Zuniga-Galindo, J.Int.Seq., 2003).

On Efficiently Solvable Cases of Quantum k-SAT

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is hbox {QMA}_1-complete (for kge 3, and where hbox {QMA}_1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

Univariate polynomials and the contractability of certain sets

We consider the set $\Pi ^*_d$ of monic polynomials $Q_d=x^d+\sum _{j=0}^{d-1}a_jx^j$, $x\in \mathbb{R}$, $a_j\in \mathbb{R}^*$, having $d$ distinct real roots, and its subsets defined by fixing the signs of the coefficients $a_j$. We show that for every choice of these signs, the corresponding subset is non-empty and contractible. A similar result holds true in the cases of polynomials $Q_d$ of even degree $d$ and having no real roots or of odd degree and having exactly one real root. For even $d$ and when $Q_d$ has exactly two real roots which are of opposite signs, the subset is contractible. For even $d$ and when $Q_d$ has two positive (resp. two negative) roots, the subset is contractible or empty. It is empty exactly when the constant term is positive, among the other even coefficients there is at least one which is negative, and all odd coefficients are positive (resp. negative).

A sequence of polynomials with optimal condition number

We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.

A New Approach to Newton-Type Polynomial Interpolation with Parameters

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

On Decoding Algebraic Codes Using Radical Locators

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an $\omega $ th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators , where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes.

Bifurcation of point vortex equilibria: four-vortex translating configurations and five-vortex stationary configurations

Point vortices in a plane form a dynamical system. A vortex equilibrium is a solution where all vortices move with a common velocity. The configuration is stationary or translating depending on whether the velocity is zero or not. In this paper, we study the bifurcation for four-vortex translating configurations and five-vortex stationary configurations. The number of distinct four-vortex translating configurations can be 0, 3, 4, 6. Among the nonzero cases, there are 1, 0, and 0, or 2 collinear ones, respectively. The number of distinct five-vortex stationary configurations can be 0, 2, 4, 6. In each case, all smaller or equal even numbers of collinear configurations among them are possible. We provide simple and clear necessary and sufficient conditions on nonzero real circulations for all the possible numbers of collinear and strictly planer equilibrium. How two or three equilibrium merge and then apart is also described. All configurations are studied through the minimal polynomial systems obtained and defined by O’Neil. The 4, 5-vortex systems contain two and three independent circulation parameters, respectively. We count the numbers of zeros by univariate polynomials after triangulating the polynomial systems. Information on the numbers of real or complex zeros and their intersection multiplicities for different components are glued together to study the bifurcation as parameters vary in and for the four-vortex and five-vortex cases, respectively. Many computations are assisted by a computer algebra system.

On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R , we consider factorizations of polynomials in R [ t ] where t is a central variable. A pseudo-root of a polynomial p ( t ) = p 0 + p 1 t + ⋯ p k t k is an element ξ ∈ R , for which there exist polynomials q 1 , q 2 such that p = q 1 ( t − ξ ) q 2 . We investigate the rational relationships that hold among the pseudo-roots of p ( t ) by using the diamond operations for cover graphs of modular lattices . When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f S that vanishes on S . By results of Leroy and Lam [16] , the set of polynomials { f T : T ⊆ S } with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Λ S of pseudo-roots of f S . Given an arbitrary subset of Λ S , our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Λ S , and in particular, rationally generates S .

Efficiently factoring polynomials modulo p4

Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg⁡f,log⁡p) time algorithm to factor a given univariate integral polynomial f modulo pk, for a prime p and k≤4.1 Thus, we solve the open question of factoring modulo p3 posed in (Sircana, ISSAC'17).Our method reduces the general problem of factoring fmodpk to that of root finding of a related polynomial E(y)mod〈pk,φ(x)ℓ〉 for some irreducible φmodp. We can efficiently solve the latter for k≤4, by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of fmodp to those modp4 (if possible). This was previously unknown, as the case of repeated factors of fmodp forbids classical Hensel lifting.

Constructing separable Arnold snakes of Morse polynomials

We give a new and constructive proof of the existence of a large class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and simple and all its critical values are distinct. Thus to any Morse polynomial we can associate an alternating permutation called Arnold snake , given by the relative positions of its critical values. Our main result is an effective construction that can realise any separable alternating permutation as the Arnold snake of a Morse polynomial.

The radial part of a class of Sobolev polynomials on the unit ball

In this work, a Sobolev inner product on the unit ball of $\mathbb {R}^{d}$ involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner product. The properties of this family constitute our main subject of study. In particular, we deduce algebraic properties, connection formulas, and some asymptotic properties. Finally, we show some numerical tests to illustrate the behavior of the roots of these univariate non-standard orthogonal polynomials.

A condition for multiplicity structure of univariate polynomials

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.