Real Polynomial Research Articles

Bohr recurrence and density of non-lacunary semigroups of ℕ

A subset R R of integers is a set of Bohr recurrence if every rotation on T d \mathbb {T}^d returns arbitrarily close to zero under some non-zero multiple of R R . We show that the set { k ! 2 m 3 n : k , m , n ∈ N } \{k!\, 2^m3^n\colon k,m,n\in \mathbb {N}\} is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if P P is a real polynomial with at least one non-constant irrational coefficient, then the set { P ( 2 m 3 n ) : m , n ∈ N } \{P(2^m3^n)\colon m,n\in \mathbb {N}\} is dense in T \mathbb {T} , thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

On the structure of the matrix in Sturm algorithm for determining the number of zeros in the open unit disk for a real polynomial

In 1984, C.W Schelin presented a numerical method for calculating the number of zeros of a real polynomial inside the unit disk using the argument principle and Cauchy index on [-1, 1] inspired by sturm’s algorithm [3], this method is based on calculation of the generalized Sturm sequence in Chebyshev form, however in some cases this algorithm does not efficient for this reason, it has been modified in another works [1], [2]. There are other numerical algorithms for calculating the number of zeros of a polynomial in the unit disk like shur -Cohn and Marden-Cohn but the numerical method presented in this study [1] is more efficient in cost of elementary operations (o (n2)). In this paper, we propose to improve this algorithm [1], to do this we study the matrix structure involved in the calculation of chebychev polynomials, for there on we prove that is a Toeplitz matrix which is known for their important properties in the complexity of calculations and we calculate its last line which is necessary to define the whole matrix to solve the linear system for calculating polynomials which are necessary to obtain Chebyshev sequence, this result will reduce calculation steps in this method [1]. This numerical algorithm may operate to study the stability of dynamical systems [2].

Quadratic Systems Possessing an Infinite Elliptic-Saddle or an Infinite Nilpotent Saddle

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems possessing exactly one elemental infinite singular point and one triple infinite singular point, which is either an infinite nilpotent elliptic-saddle or a nilpotent saddle. This class can be divided into three different families, namely, [Formula: see text] of phase portraits possessing three real finite singular points, [Formula: see text] of phase portraits possessing one real and two complex finite singular points, and [Formula: see text] of phase portraits possessing one real triple finite singular point. Here, we provide a comprehensive study of the geometry of these three families. Modulo the action of the affine group and time homotheties, families [Formula: see text] and [Formula: see text] are three-dimensional and family [Formula: see text] is two-dimensional. We study the respective bifurcation diagrams of their closures with respect to specific normal forms, in sub-sets of real Euclidean spaces. The bifurcation diagram of family [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) yields 1274 (resp., 89 and 14) sub-sets with 91 (resp., 27 and 12) topologically distinct phase portraits for systems in the closure [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) within the representatives of [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) given by a specific normal form.

Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert's 16th Problem.

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of |R| to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n≥2 have the maximum number of 2(n-1)(4(n-1)-2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

Approximation properties by shifted knots type of α-Bernstein–Kantorovich–Stancu operators

Through the real polynomials of the shifted knots, the α-Bernstein–Kantorovich operators are studied in their Stancu form, and the approximation properties are obtained. We obtain some direct approximation theorem in terms of Lipschitz type maximum function and Peetre’s K-functional, as well as Korovkin’s theorem. Eventually, the modulus of continuity is used to compute the upper bound error estimation.

Structured ramp secret sharing schemata over rings of real polynomials

Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ⩾2, groups, that are specified by the dealer's levels Li for i=1,2,…,ℓ and each level Li, i⩾2, is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach.

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

Schinzel-type bounds for the Mahler measure on lemniscates

We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials that achieve our lower bound for the generalized Mahler measure. It turns out that the extremal polynomials attaining the bound must have even degree. The main computational part of this work is related to finding many extremals of degree four and higher, which is a new feature compared to the original Schinzel’s theorem where only quadratic irreducible extremals are possible.

Two-Point Polynomial Patterns in Subsets of Positive Density in ℝn

Abstract Let $\gamma (t)=(P_{1}(t),\ldots ,P_{n}(t))$ where $P_{i}$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma (t)\}$ in sets of positive density $\epsilon $ in $[0,1]^{n}$ with a gap estimate $t\geq \delta (\epsilon )$ when $P_{i}$’s are arbitrary, and in $[0,N]^{n}$ with a gap estimate $t\geq \delta (\epsilon )N^{n}$ when $P_{i}$’s are of distinct degrees where $\delta (\epsilon )=\exp \left (-\exp \left (c\epsilon ^{-4}\right )\right )$ and $c$ only depends on $\gamma $. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain’s reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.

Maximum number of limit cycles of a second order differential system

In this paper, we study the limit cycles of a perturbed differential in $\mathbb{R} ^2$, given by \begin{equation*} \left\{ \begin{array}{ccl} \overset{.}{x} y ,\\ \overset{.}{y} -x-\epsilon (1+\sin^n (\theta) \cos^m (\theta))H(x,y), \end{array} \right. \end{equation*} where $\epsilon$ is a small parameter, $m$ and $n$ are non-negative integers, $\tan(\theta)=y/x$, and $P(x,y)$ is a real polynomial of degree $n \geq 1$. Using Averaging theory of first order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to confirm and illustrate our results.

Maximal and typical topology of real polynomial singularities

Maximal and typical topology of real polynomial singularities

Modeling and mathematical approach for mechanical natural frequencies of structural plates with free edges

The current study aims to demonstrate an easy mathematical procedure and simple to be used by engineers to maintain the mechanical natural frequencies (Eigen values of free vibration) of thin rectangular structural plates with free edges included in boundary conditions. The reason for this easiness and simplicity is that new method of analysis depends mainly on using mechanical minimum energy concept represented in the Rayleigh–Ritz method with appropriate real polynomials as admissible functions. The new procedure guides these functions to satisfy the boundary conditions at the middle points on the free edges of the plate beside the nodal points of the vibration modes along each edge direction. Therefore, there are no complicated or imaginary terms in the developed functions while they lead to surprisingly acceptable and even accurate results when compared with other works and Ansys program models.

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

A new algorithm for computing the nearest polynomial to multiple given polynomials via weighted ℓ2,q-norm minimization and its complex extension

A new algorithm is proposed in this paper for computing the nearest polynomial to multiple given polynomials with a given zero in the real case, where the distance between polynomials is defined by the weighted ℓ2,q norm (0<q≤2). First, the problem is formulated as a univariate constrained non-convex minimization problem, where prior information of the coefficients of the polynomial can be embedded by selecting proper weights. Then, an iteratively reweighted algorithm is designed to solve the obtained problem, and also the convergence and rate of convergence are uniformly demonstrated for all q in (0,2]. Since all the existing methods for computing the nearest polynomial to multiple given polynomials with a given zero are limited to the real case, we ingeniously extend the results to the complex case. Finally, two representative examples that separately compute the nearest real and complex polynomials are presented to show the effectiveness of the proposed algorithm.

In search of Newton-type inequalities

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.

Efficient quantum amplitude encoding of polynomial functions

Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the application of these algorithms. Here, we present and compare two efficient methods for the amplitude encoding of real polynomial functions on n qubits. This case holds special relevance, as any continuous function on a closed interval can be uniformly approximated with arbitrary precision by a polynomial function. The first approach relies on the matrix product state representation (MPS). We study and benchmark the approximations of the target state when the bond dimension is assumed to be small. The second algorithm combines two subroutines. Initially we encode the linear function into the quantum registers either via its MPS or with a shallow sequence of multi-controlled gates that loads the linear function's Hadamard-Walsh series, and we explore how truncating the Hadamard-Walsh series of the linear function affects the final fidelity. Applying the inverse discrete Hadamard-Walsh transform converts the state encoding the series coefficients into an amplitude encoding of the linear function. Thus, we use this construction as a building block to achieve an exact block encoding of the amplitudes corresponding to the linear function on k0 qubits and apply the quantum singular value transformation that implements a polynomial transformation to the block encoding of the amplitudes. This unitary together with the Amplitude Amplification algorithm will enable us to prepare the quantum state that encodes the polynomial function on k0 qubits. Finally we pad n&amp;#x2212;k0 qubits to generate an approximated encoding of the polynomial on n qubits, analyzing the error depending on k0. In this regard, our methodology proposes a method to improve the state-of-the-art complexity by introducing controllable errors.

Real univariate polynomials with given signs of coefficients and simple real roots

We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\leq d$);2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$ having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.

Determinantal Representations and the Image of the Principal Minor Map

Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $\mathbb{F}$, there is no finite set of equations whose orbit under $(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$ matrices over ${\mathbb{F}}$ under the principal minor map for every $n$.

Real stable polynomials and the alternatingly increasing property

In this paper, we first present a criterion for the real stability of recursive polynomials by applying the characterization of Borcea and Brändén concerning linear operators preserving stability. As applications, we derive some real stability results occurred in the literature in a unified manner. Moreover, we prove the real-rootedness of peak polynomials in bicolored dyck paths and ternary polynomials.Second, based on the h-polynomials from combinatorial geometry, we use real stability to get two criteria for interlacing property of zeros between a polynomial and its reciprocal polynomial, which in particular imply the alternatingly increasing property of the original polynomial or its reciprocal polynomial. These criteria extend a result of Brändén and Solus and unify such properties for many combinatorial polynomials, including ascent polynomials for k-ary words, descent polynomials on signed Stirling permutations and colored permutations, q-analog of descent polynomials on colored permutations and descent polynomials on color-signed permutations. Furthermore, we also obtain a recurrence relation and zero-interlacing property of q-analog of descent polynomials on colored permutations that extend some results of Brändén and Brenti. In addition, as an application of weak Hurwitz stability, we get the alternatingly increasing property and zero-interlacing property for two kinds of peak polynomials on the dual set of Stirling permutations.