Original Polynomial Research Articles

On the reduction of matrix polynomials to Hessenberg form

It is well known that every real or complex square matrix is unitarily similar to an upper Hessenberg matrix. The purpose of this paper is to provide a constructive proof of the result that every square matrix polynomial can be reduced to an upper Hessenberg matrix, whose entries are rational functions and in special cases polynomials. It will be shown that the determinant is preserved under this transformation, and both the finite and infinite eigenvalues of the original matrix polynomial can be obtained from the upper Hessenberg matrix.

Constructing strong ℓ-ifications from dual minimal bases

We provide an algorithm for constructing strong ℓ-ifications of a given matrix polynomial P(λ) of degree d and size m×n using only the coefficients of the polynomial and the solution of linear systems of equations. A strong ℓ-ification of P(λ) is a matrix polynomial of degree ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial P(λ). All explicit constructions of strong ℓ-ifications introduced so far in the literature have been limited to the case where ℓ divides d, though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong ℓ-ifications for wider choices of the degree ℓ, namely, when ℓ divides one of nd or md (and d≥ℓ). In the case where ℓ divides nd (respectively, md), the strong ℓ-ifications we construct allow us to easily recover the minimal indices of P(λ). In particular, we show that they preserve the left (resp., right) minimal indices of P(λ), and the right (resp., left) minimal indices of the ℓ-ification are the ones of P(λ) increased by d−ℓ (each). Moreover, in the particular case ℓ divides d, the new method provides a companion ℓ-ification that resembles very much the companion ℓ-ifications already known in the literature.

Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation

The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been completely analysed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated to dispersal is either beneficial or detrimental to total population abundance. Therefore, this is a contribution to the SLOSS question. Importantly, we also show that, depending on the underlying mechanism, there is no unique way to generalize the logistic model to a patchy situation. In many cases, the standard model is not the correct generalization. We analyse several alternative models and compare their predictions. Finally, we emphasize the shortcomings of the logistic model when written in the r-K parameterization and we explain why Verhulst’s original polynomial expression is to be preferred.

Enhance fuzzy vault security using nonrandom chaff point generator

Toward the combination of cryptographic and biometric systems, by performing specific binding technique on cryptographic key and biometric template, the fuzzy vault framework enhances security level of current biometric cryptographic systems in terms of hiding secret key and protecting the template. Though original scheme suggests the use of error-correction techniques (e.g., Reed-Solomon code) to reconstruct the original polynomial, recent implementations do not share the same point of view. Instead, Cyclic Redundant Code (CRC) is applied to identify the genuine polynomial from set of candidates due to its simplicity. Within the scope of this article, we address a significant flaw of current CRC-based fuzzy vault schemes, which allows the potential of successful blend substitutions attack. To overcome that problem, an integration of two novel modules into general fuzzy vault scheme, namely chaff points generator and verifier, are proposed. The new modules are designed to be integrated easily into existing systems as well as simple to enhance. The proposed scheme can detect any modification in vault and, as a result, eliminate the blend substitutions attack to improve general security. Moreover, our experimental results with real-world datasets show an increasing of Genuine Acceptance Rates (GAR).

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

Analogies between gravitation and electromagnetism have been known since the 1950s. Here, we examine a fairly general type D solution — the exact seven parameter solution of Plebański–Demiański (PD) — to demonstrate these analogies for a physically meaningful spacetime. The two quadratic curvature invariants B2 - E2 and E⋅B are evaluated analytically. In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitoelectric mass and gravitoelectric current of the PD solution, respectively, if there are no gravitomagnetic monopoles present. Furthermore, the square of the Bel–Robinson tensor reads (B2 + E2)2 for the PD solution, reminiscent of the square of the energy density in electrodynamics. By analogy to the energy–momentum 3-form of the electromagnetic field, we provide an alternative way to derive the recently introduced Bel–Robinson 3-form, from which the Bel–Robinson tensor can be calculated. We also determine the Kummer tensor, a tensor cubic in curvature, for a general type D solution for the first time, and calculate the pieces of its irreducible decomposition. The calculations are carried out in two coordinate systems: In the original polynomial PD coordinates and in a modified Boyer–Lindquist-like version introduced by Griffiths and Podolský (GP) allowing for a more straightforward physical interpretation of the free parameters.

Every discrete 2D autonomous system admits a finite union of parallel lines as a characteristic set

In this paper, we show that every discrete 2D autonomous system, that is described by a set of linear partial difference equations with constant real coefficients, admits a finite union of parallel lines as a characteristic set. In order to prove our claim, we first look at a special class of scalar discrete 2D systems and provide such characteristic sets for systems in this class. This special class has the property that systems in this class have their quotient rings to be finitely generated modules over a one-variable Laurent polynomial subring of the original two-variable Laurent polynomial ring in the shift operators. We show that such systems always admit a finite collection of horizontal lines for a characteristic set. We then extend this result to non-scalar discrete 2D autonomous systems. We achieve this in two steps: first, we show that every scalar discrete 2D system can be converted into a system in the above-mentioned class by a coordinate transformation on the independent variables set, $$\mathbb {Z}^2$$Z2. Using this we then show that characteristic sets for the original system can be found by applying the inverse coordinate transformation on characteristic sets of the transformed system. Since the transformed system, by virtue of being in the special class, admits a finite union of horizontal lines as a characteristic set, the original system is guaranteed to admit a characteristic set that is a coordinate transformation applied to a finite union of horizontal lines. The coordinate transformation maps this union of horizontal lines to a union of parallel, but possibly tilted, lines. In the next step, we generalize the scalar case to the general vector case: that is, systems with more than one dependent variables. The main motivation for studying characteristic sets that are unions of finitely many parallel lines is that, arguably, such sets can be called "thin" in $$\mathbb {Z}^2$$Z2 in comparison to the prevalent notions of convex cones and half-spaces as characteristic sets (see "Appendix 1").

Smaller SDP for SOS decomposition

A popular numerical method to compute sum of squares (SOS of polynomials) decompositions for polynomials is to transform the problem into semi-definite programming (SDP) problems and then solve them by SDP solvers. In this paper, we focus on reducing the sizes of inputs to SDP solvers to improve the efficiency and reliability of those SDP based methods. Two types of polynomials, convex cover polynomials and split polynomials, are defined. A convex cover polynomial or a split polynomial can be decomposed into several smaller sub-polynomials such that the original polynomial is SOS if and only if the sub-polynomials are all SOS. Thus the original SOS problem can be decomposed equivalently into smaller sub-problems. It is proved that convex cover polynomials are split polynomials and it is quite possible that sparse polynomials with many variables are split polynomials, which can be efficiently detected in practice. Some necessary conditions for polynomials to be SOS are also given, which can help refute quickly those polynomials which have no SOS representations so that SDP solvers are not called in this case. All the new results lead to a new SDP based method to compute SOS decompositions, which improves this kind of methods by passing smaller inputs to SDP solvers in some cases. Experiments show that the number of monomials obtained by our program is often smaller than that by other SDP based software, especially for polynomials with many variables and high degrees. Numerical results on various tests are reported to show the performance of our program.

Improvement of FPPR method to solve ECDLP

Solving the elliptic curve discrete logarithm problem (ECDLP) by using Grobner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugere, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough.

Rational sums of hermitian squares of free noncommutative polynomials

In this paper we consider polynomials in noncommuting variables that admit sum of hermitian squares and commutators decompositions. We recall algorithms for finding decompositions of this type that are based on semidefinite programming. The main part of the article investigates how to find such decomposition with rational coefficients if the original polynomial has rational coefficients. We show that the numerical evidence, obtained by the Gram matrix method and semidefinite programming, which is usually an almost feasible point, can be frequently tweaked to obtain an exact certificate using rational numbers. In the presence of Slater points, the Peyrl-Parrilo rounding and projecting method applies. On the other hand, in the absence of strict feasibility, a variant of the facial reduction is proposed to reduce the size of the semidefinite program and to enforce the existence of Slater points. All these methods are implemented in our open source computer algebra package NCSOStools. Throughout the paper many worked out examples are presented to illustrate our results.

Resolution enhancement in second-derivative spectra.

Derivative spectroscopy is a powerful tool for the resolution enhancement in infrared, near-infrared, Raman, ultraviolet-visible, nuclear magnetic resonance, electron paramagnetic resonance, and fluorescence spectroscopy. Despite its great significance in analytical chemistry, not all aspects of the applications of this method have been explored as yet. This is the first systematic study of the parameters that influence the resolution enhancement in the second derivative spectra. The derivative spectra were calculated with the Savitzky-Golay method with different window size (5, 15, 25) and polynomial order (2, 4). The results obtained in this work show that the resolution enhancement in the second derivative spectra strongly depends on the data spacing in the original spectra, window size, polynomial order, and peak profile. As shown, the resolution enhancement is related to variations in the width of the peaks upon the differentiation. The present study reveals that in order to maximize the separation of the peaks in the second derivative spectra, the original spectra should be recorded at high resolution and differentiated using a small window size and high polynomial order. However, working with the real spectra one has to compromise between the noise reduction and optimization of the resolution enhancement in the second derivative spectra.

Linearization of Lagrange and Hermite interpolating matrix polynomials

This paper considers interpolating matrix polynomials P (λ) in Lagrange and Hermite bases. A classical approach to investigate the polynomial eigenvalue problem P (λ)x = 0 is linearization, by which the polynomial is converted into a larger matrix pencil with the same eigenvalues. Since the current linearizations of degree n Lagrange polynomials consist of matrix pencils with n + 2 blocks, they introduce additional eigenvalues at infinity. Therefore, we introduce new linearizations which overcome this. Initially, we restrict to Lagrange and barycentric Lagrange matrix polynomials and give two new and more compact linearizations, resulting in matrix pencils of n+ 1 and n blocks for polynomials of degree n. For the latter, there is a one-to-one correspondence between the eigenpairs of P (λ) and the eigenpairs of the pencil. We also prove that these linearizations are strong. Moreover, we show how to exploit the structure of the proposed matrix pencils in Krylov-type methods, so that in this case we only have to deal with linear system solves of matrices of the original matrix polynomial dimension. Finally, we generalize for multiple interpolation and introduce new linearizations for Hermite Lagrange and barycentric Hermite matrix polynomials. Again, we can show that the linearizations are strong and that there is a one-to-one correspondence of the eigenpairs.

An improved bounded semi-Lagrangian scheme for the turbulent transport of passive scalars

An improved bounded semi-Lagrangian scalar transport scheme based on cubic Hermite polynomial reconstruction is proposed in this paper. Boundedness of the scalar being transported is ensured by applying derivative limiting techniques. Single sub-cell extrema are allowed to exist as they are often physical, and help minimize numerical dissipation. This treatment is distinct from enforcing strict monotonicity as done by D.L. Williamson and P.J. Rasch [5], and allows better preservation of small scale structures in turbulent simulations. The proposed bounding algorithm, although a seemingly subtle difference from strict monotonicity enforcement, is shown to result in significant performance gain in laminar cases, and in three-dimensional turbulent mixing layers. The scheme satisfies several important properties, including boundedness, low numerical diffusion, and high accuracy. Performance gain in the turbulent case is assessed by comparing scalar energy and dissipation spectra produced by several bounded and unbounded schemes. The results indicate that the proposed scheme is capable of furnishing extremely accurate results, with less severe resolution requirements than all the other bounded schemes tested. Additional simulations in homogeneous isotropic turbulence, with scalar timestep size unconstrained by the CFL number, show good agreement with spectral scheme results available in the literature. Detailed analytical examination of gain and phase error characteristics of the original cubic Hermite polynomial is also included, and points to dissipation and dispersion characteristics comparable to, or better than, those of a fifth order upwind Eulerian scheme.

On the inverse eigenvalue problem for T-alternating and T-palindromic matrix polynomials

The inverse eigenvalue problem for T-alternating matrix polynomials over arbitrary algebraically closed fields of characteristic different from two is considered. The main result shows that the necessary conditions obtained in [9] for a matrix polynomial to be the Smith form of a T-alternating matrix polynomial are under mild conditions also sufficient to be the Smith form of a T-alternating matrix polynomial with invertible leading coefficient which is additionally in anti-triangular form. In particular, this result implies that any T-alternating matrix polynomial with invertible leading coefficient is equivalent to a T-alternating matrix polynomial in anti-triangular form that has the same finite and infinite elementary divisors as the original matrix polynomial. Finally, the inverse eigenvalue problem for T-palindromic matrix polynomials is considered excluding the case that both +1 and −1 are eigenvalues.

Blending of the Trigonometric Polynomial Spline Curve with Arbitrary Continuous Orders

In order to develop the theory of trigonometric polynomial spline curve, the representation of trigonometric polynomial spline curve is blended to a general form based on blending of trigonometric polynomial curves. Moreover, some properties of the blending curve are discussed in details. The research shows that the basis of the trigonometric polynomial curve is relative simple, and the blending curve includes the original trigonometric polynomial spline curve and shows much better shape-control capability than the original curve. Meanwhile, the blending curve keeps the same degree of original one. It is easy to find that the curve can be reshaping by adjusting the shape factor. At the same time, a new method of the representation of closed curve is given and can also accurately represent the whole ellipse arc and other conic curve.

On the Tutte–Krushkal–Renardy polynomial for cell complexes

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte–Krushkal–Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte–Krushkal–Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte–Krushkal–Renardy polynomial.

The Mother Minkowski Algebra of Order m

It is found that all polynomials of up to degree m have an encoding as m-vectors in a geometric algebra referred to as the Mother Minkowski algebra of order m. It is then shown that all conformal transformations may be applied to these m-vectors, the results of which, when converted back into polynomial form, give us the transformed surfaces in terms of the zero sets of the original and final polynomials.

Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials

Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

A bootstrapping approach for computing multiple solutions of differential equations

Discretizing systems of nonlinear algebraic differential equations yields polynomial systems. When using a fine discretization, the resulting polynomial system is often too large to solve using a direct solving approach. Our approach for solving such systems is to utilize a homotopy continuation based method arising from domain decomposition. This method solves polynomial systems arising from subdomains and then uses homotopy continuation to build solutions of the original polynomial system. We illustrate this approach on both one- and two-dimensional problems.

An efficient reliable algorithm for the approximation of all polynomial roots based on the method of D. Bernoulli

Specifications. Let Jm := {1, . . . ,m} for m ∈ N1 := {1, 2, . . . } and let z1, . . . , zk, k ∈ N1∖{1}, be complex numbers with zg = zh for g, h ∈ Jk, g = h, and 0 < |z1| 6 · · · 6 |zk|. A polynomial p(z) is given by p(z) = ∑m i=0 aiz i with ai ∈ C, i ∈ Jm, am = 0 and a0 = −1. It denotes either the original polynomial, a shift of it or a polynomial obtained by deflation. In the product representation p(z) = am ∏k j=1(z− zj)j with mj ∈ N1 the roots z1, . . . , zk are complex numbers with properties specified above. If not stated otherwise, the index j ranges in Jk, and the index n is in N0 := N1 ∪ {0}. For example, (qn)n denotes the sequence (qn)n∈N0 . Throughout this paper the following abbreviations are used: