Multivariate Polynomial Equations Research Articles

Counting \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 black holes as algebraic varieties

We calculate the helicity trace index B14 for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations modulo gauge symmetries. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. The first three methods rely on a mixture of symbolic and high precision numerics whereas the Hilbert series is symbolic and admit a gauge invariant analysis. Furthermore, exploiting various exchange symmetries, we show that quartic and higher order terms are absent in the potential, which if present would have spoiled the counting. Incorporating recent developments in algebraic geometry focusing on computational algorithms, we have extended the scope of one of the authors previous works [1, 2] and presented a new perspective for the black hole microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 2 black hole microstate counting.

On the existence, uniqueness and calculation of Wiener path integral most probable paths for determining the stochastic response of nonlinear dynamical systems

Various techniques are developed for addressing the existence, uniqueness and numerical calculation of Wiener path integral (WPI) most probable path solutions. Specifically, the WPI technique for determining the stochastic response of diverse nonlinear dynamical systems treats the system response joint transition probability density function as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. This functional integral is evaluated, ordinarily, by resorting to an approximate approach that considers the contribution only of the most probable path. The most probable path corresponds to an extremum of the functional integrand and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP).In this paper, first, it is shown that for the commonly considered case of the system nonlinearity being of polynomial form, there exist globally optimal solutions corresponding to the most probable path BVP. Further, relying on algebraic geometry concepts and tools, a condition is derived for determining if the BVP for the most probable path exhibits a unique solution over a specific region. Furthermore, a novel solution approach is developed for the BVP by relying on Sylvester’s dialytic method of elimination. Notably, the method reduces the complexity of the BVP system of coupled multivariate polynomial equations by eliminating one or more variables. Various numerical examples pertaining to diverse nonlinear oscillators are included for demonstrating the capabilities of the developed techniques.

Progress in Multivariate Cryptography: Systematic Review, Challenges, and Research Directions

Multivariate Public Key Cryptosystem (MPKC) seem to be promising toward future digital security even in the presence of quantum adversaries. MPKCs derive their security from the difficulty of solving a random system of multivariate polynomial equations over a finite field, which is known to be an NP-hard problem. This article aims at presenting a comprehensive survey that covers multivariate public key encryption and signature schemes specifically targeting toward security, efficiency, and parameter choice. The survey starts by giving an overview of the existing security challenges which include structural attacks such as MinRank attack, differential attack , and finding Gröbner basis for direct attack , and so on. Additionally, it discusses the necessary algorithms for the implementation of the multivariate schemes. This study also compares the promising multivariate encryption and signature schemes. The critical open challenges that are reviewed in this survey will serve as a single comprehensive source of information on multivariate encryption and signature schemes and a ready reference for researchers working in this rising area of public key cryptography.

Selection Strategy of F4-Style Algorithm to Solve MQ Problems Related to MPKC

Multivariate public-key cryptosystems are potential candidates for post-quantum cryptography. The security of multivariate public-key cryptosystems relies on the hardness of solving a system of multivariate quadratic polynomial equations. Faugère’s F4 algorithm is one of the solution techniques based on the theory of Gröbner bases and selects critical pairs to compose the Macaulay matrix. Reducing the matrix size is essential. Previous research has not fully examined how many critical pairs it takes to reduce to zero when echelonizing the Macaulay matrix in rows. Ito et al. (2021) proposed a new critical-pair selection strategy for solving multivariate quadratic problems associated with encryption schemes. Instead, this paper extends their selection strategy for solving the problems associated with digital signature schemes. Using the OpenF4 library, we compare the software performance between the integrated F4-style algorithm of the proposed methods and the original F4-style algorithm. Our experimental results demonstrate that the proposed methods can reduce the processing time of the F4-style algorithm by up to a factor of about seven under certain specific parameters. Moreover, we compute the minimum number of critical pairs to reduce to zero and propose their extrapolation outside our experimental scope for further research.

Patterns of Influence of Meteorological Elements on Maize Grain Weight and Nutritional Quality

Meteorological factors are one of the important factors that determine maize kernel weight and grain nutritional quality. Analyzing the influence of meteorological factors before and after anthesis on maize kernel weight and nutritional quality components is of great significance for improving corn yield and quality. Therefore, five different maize hybrids and conducted continuous experiments from 2018 to 2021 were selected in this study, to explore the response of maize kernel weight and grain nutritional quality to meteorological factors in different growth periods, and to quantify the linear relationship between grain nutritional quality parameters, grain weight, and meteorological factors. The main results were as follows: the 100-grain weight reached the maximum value of 39.53 g in 2018; the contents of crude protein, total starch, and crude fat in grains reached the maximum in 2018, 2020, and 2018, respectively, which were 9.61%, 69.2%, and 5.1%. Meteorological factors significantly affected the maize grain weight (p < 0.05). Before anthesis, total sunshine duration, average temperature, relative humidity, and the accumulated temperature had strong effects on grain weight. After anthesis, average daily temperature, total rainfall, temperature difference, accumulated temperature, average daily highest temperature, and total sunshine hours had strong effects on grain weight. There was also a significant correlation between grain weight and grain nutritional quality components (p < 0.05). The multivariate polynomial equation analysis revealed that further potential for maize grain weight could be exploited by adjusting the content of each quality component of the kernels under the current test conditions. Meteorological elements can indirectly affect the 100-grain weight through their relationship with the nutritional quality of the grains, with accumulated temperature before anthesis, average temperature after anthesis, and accumulated temperature after anthesis having the greatest indirect effect on the 100-grain weight. Therefore, the effects of pre-anthesis accumulation temperature, post-anthesis average temperature, and post-anthesis accumulation temperature on the nutritional quality of the grains can be harmonized by the application of hybrids of different lengths of vegetation and by adjusting the sowing time in agricultural production. Ultimately, maize grain weight can be increased on the basis of optimizing the content of various quality components in the grains.

The complexity of solving Weil restriction systems

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Gröbner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension k↪K, associates to a system F defined over K a system Weil(F) defined over k, in such a way that the solutions of F over K and those of Weil(F) over k are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system Weil(F) obtained via Weil restriction in terms of algebraic invariants of the system F.

Design of higher order iteration maps for multi-variable polynomials via algebraic geometry

In this paper, algorithmic procedures based on algebraic geometry tools are proposed to design iteration maps with arbitrary order of superlinear convergence, for the solution of systems of multi-variable polynomial equations. First, the design is carried out in the single-variable case to illustrate its properties and to highlight its relation with the celebrated Newton and Householder iterative methods. Secondly, the proposed techniques are extended to the multi-variable case. The effectiveness of the proposed approach is highlighted via its application to the inverse kinematics of a robot arm.

Enumeration of labelled 4-regular planar graphs II: Asymptotics

This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number gn of labelled 4-regular planar graphs on n vertices. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

A NOTE ON THE SOLUTION OF THE CHARACTERISTIC EQUATION OVER THE SYMMETRIZED MAX-PLUS ALGEBRA

The symmetrized max-plus algebra is an extension of max-plus algebra. One of the problems in the symmetrized max-plus algebra is determining the eigenvalues of a matrix. If the determinant can be defined, the characteristic equation can be formulated as a max-plus algebraic multivariate polynomial equation system. A mathematical tool for solving the problem using operations as in conventional algebra, known as the extended linear complementary problem (ELCP), to determine the solution to the characteristic equation. In this paper, we will describe the use of ELCP in determining the solution to the characteristic equations of matrices over the symmetrized max-plus algebra.

On generalizing Descartes' rule of signs to hypersurfaces

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.

An algorithm for simultaneous nonsymmetric algebraic Riccati equations over finite fields

An algorithm is proposed for the solution of the NP-hard problem of simultaneous nonsymmetric algebraic Riccati equations over finite fields. The proposed algorithm has an application to the solution of some structured quadratic polynomial equations over finite fields with complexity of O˜2(nℓmax+n+ℓmax) for m⋅n variables and m⋅n equations over F2, where n+ℓmax (ℓmax≤m and m arbitrary) is the dimension of a largest invariant subspace (of some special type) of the m+n×m+n matrix of coefficients of a single Riccati equation. The method can be used to attack or crypt-analyze cryptographic systems that are based on structured multivariable quadratic polynomial equations.

Determination of Discharge Coefficient for both Semicircular and Triangular Labyrinth Weirs

A Labyrinth weir is a type of nonlinear weir folded in plan-view, with the ability to pass a large flow at low heads due to an increase in the effective length of the weir crest for a given channel width. The discharge coefficients (Cd) were experimentally determined for both semicircular and triangular labyrinth weirs of varying sidewall angles (α) under free-flow conditions using eight physical models. Dimensional analyses are conducted using the Buckingham π theorem to find the dependent and non-dependent variables. A nonlinear empirical equation was developed using (Statistical Package for the Social Sciences 22) software. The findings show that the discharge coefficient decreases with the increase of headwater to (Ht/P) for both types of labyrinth weir with different plan forms. It is found that the efficiency of the semicircular and triangular labyrinth is higher than linear weir for (Ht/P ≤ 0.45) and decreases with increasing the value of (Ht/P) due to interference of flow between the layers in downstream. The obtained results revealed that the discharge coefficient (Cd) values obtained empirically using multivariable power regression and 5th degree polynomial equations considerably agree with the experimental data, for both of the semicircular and triangular labyrinth weirs. Furthermore, the correlation coefficient R2was (0.9951) and (0.9637) using 5th degree polynomial equation, and their values were (0.9643) and (0.9274) using multivariable power regression equation for both semicircular and triangular labyrinth weir models, respectively. The results in the present study compared quite well with the predicted results of the proposed equations obtained by other investigators.

An improvement of a key exchange protocol relying on polynomial maps

Akiyama et al. [A key exchange protocol relying on polynomial maps, Int. J. Math. Ind. 11 (2019) 195003] proposed a post-quantum key exchange protocol that is based on the hardness of solving a system of multivariate nonlinear polynomial equations but has a design strategy different from ordinary multivariate cryptography. Their protocol has two versions, an original one and a modified one, where the modified one has a trade-off that its security is strengthened while it has nonzero error probability in establishing a common key. In fact, the evaluation in their paper suggests that the probability of failing to establish a common key by the modified protocol with the proposed parameter set is impractically high. In this paper, we improve the success probability of Akiyama et al.’s modified key exchange protocol significantly while keeping the security, by restricting each component of the correct common key from the whole of the coefficient field to its small subset. We give theoretical and experimental evaluations showing that our proposed parameter set for our protocol is expected to achieve both failure probability [Formula: see text] and 128-bit security level.

A computational algebraic geometry technique for determining nonlinear normal modes of structural systems

A computational algebraic geometry technique is developed for determining nonlinear normal modes (NNMs) of multi-degree-of-freedom (MDOF) nonlinear dynamic systems. Specifically, resorting to computational algebraic geometry concepts and tools, such as Gröbner basis, it is shown that the complete solution set of a nonlinear algebraic system of coupled multivariate polynomial equations is determined exactly, even for a relatively large number of unknowns. Herein, these unknowns represent expansion coefficients for approximating NNMs of nonlinear dynamic systems. In this regard, the exact nature of the determined coefficient vector, which is one the significant advantages of the technique, implies increased confidence in approximating NNMs more accurately by utilizing higher-order expansions with additional terms. Several numerical examples referring to various MDOF nonlinear dynamic systems are considered for demonstrating the efficacy and potential of the solution technique. Also, note that although the technique is developed herein by relying on the classical Shaw–Pierre definition of NNMs, this is not necessarily a restriction of the methodology. In fact, the proposed computational algebraic geometry solution framework can be potentially used, in a rather straightforward manner, in conjunction with alternative NNM definitions and problem formulations requiring the solution of nonlinear algebraic equations of polynomial form.

English

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and Buchberger's algorithm, are superexponential. The superexponential complexity of the well-known algorithms is due to their not being "minimal" in a certain sense. Buchberger's algorithm produces a Groebner basis. The proposed original reduction algorithm achieves the required task, via computation of determinants of parametric Sylvester matrices, and produces a Rabin basis, which is shown to be minimal, when two multivariate polynomials are reduced at a time. The minimality of Rabin basis allows us to prove exponential lower bounds for the space complexity of an algebraic proof of certification, for a specific computational problem in the computational complexity class PSPACE, showing that the complexity classes PSPACE and P cannot be the same. By the same reasoning, it follows that Co-NP is not the same as either of the complexity classes NP and P, and that the polynomial time hierarchy does not collapse. It is also shown that the class BPP of languages decidable by bounded error probabilistic algorithms with (probabilistic) polynomial time proofs for the membership of input words is not the same as any one of the complexity classes P, NP and Co-NP. It follows again, from the discussions, that the complexity classes NP and P are not the same, by relativization of BPP, with respect to P and NP.

A note on the univariate nonic derived from the coupler curve of four-bar linkages

The coupler curve is the curve traced by one of the points on the coupler link of a four-bar linkage, which is algebraically a sextic bivariate polynomial. A system of eighth-order polynomial equations of nine unknowns can be established from the algebraic coupler curve in order to determine linkage parameters. However, such a system of higher order multivariate polynomial equations poses a big challenge to find solutions efficiently. In this paper, a univariate 9th order polynomial is derived from the system of higher order polynomials. The derivation of the univariate polynomial is presented in details. Examples of coupler curve synthesis are included to show the efficient solutions with the univariate polynomial.

A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations

We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.

Globally Optimal H2-Norm Model Reduction: A Numerical Linear Algebra Approach

We show that the H2-norm model reduction problem for single-input/single-output (SISO) linear time-invariant (LTI) systems is essentially an eigenvalue problem (EP), from which the globally optimal solution(s) can be retrieved. The first-order optimality conditions of this model reduction problem constitute a system of multivariate polynomial equations that can be converted to an affine (or inhomogeneous) multiparameter eigenvalue problem (AMEP). We solve this AMEP by using the so-called augmented block Macaulay matrix, which is introduced in this paper and has a special (block) multi-shift invariant null space. The set of all stationary points of the optimization problem, i.e., the (2r)-tuples (r is the order of the reduced model) of affine eigenvalues and eigenvectors of the AMEP, follows from a standard EP related to the structure of that null space. At least one of these (2r)-tuples corresponds to the globally optimal solution of the H2-norm model reduction problem. We present a simple numerical example to illustrate our approach.

Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity

At PKC 2019, Clear and McGoldrick presented the first identity-based encryption (IBE) scheme that is group homomorphic for addition modulo a poly-sized prime e. Assuming that deciding solvability of a special system of multivariate polynomial equations is hard, they proved that their scheme for $$e>2$$ is anonymous. In this paper, we review the classical Galbraith’s test on the anonymity of the first pairing-free IBE scheme due to Cocks. With the eye of the reciprocity law for $$\mathbb {F}_\mathtt {q}[x]$$ , we can have a profound understanding of the test and naturally extend it to give a practical attack on the anonymity of the Clear–McGoldrick IBE scheme. Furthermore, we believe that our technique plays a crucial role in anonymizing IBE schemes from higher residuosity.

Modified Public Key Cryptosystem Based On Circulant Matrix

Experts believe that public key cryptosystems on non-commutative algebraic structures are resistant to the attack of quantum algorithms. In recent years, public key cryptosystems based on Polynomial Symmetrical Decomposition (PSD) on the non-commutative group have been developed. However, they are vulnerable to direct attack, linearization equations attack, and overdefined systems of multivariate polynomial equations attack. This cryptosystem has also been improved by experts. However, the operation of the proposed PSD Improvement still uses complex computing and untested. Therefore in this paper, we replace PSD on a non-commutative group into a non-commutative matrix group. We chose the circulant matrix on the key agreement protocol and the key distribution. The results show that the cryptosystem proposed on the circulant matrix is resistant to direct attack, linearization equations attack, and overdefined systems of multivariate polynomial equations attack.