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Classically, the Bezout matrix or simply Bezoutian of two polynomials is used to locate the roots of the polynomial and, in particular, test for stability. In this paper, we develop the theory of Bezoutians on real Riemann surfaces of dividing type. The main result connects the signature of the Bezoutian of two real meromorphic functions to the topological data of their quotient, which can be seen as the generalization of the classical Cauchy index. As an application, we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the Bezoutian. We provide examples of our method in the case of simply connected quadrature domains. [ABSTRACT FROM AUTHOR]

In this paper we construct HNN-extensions of diassociative algebras and Leibniz algebras and prove that every Leibniz algebra embeds into any of its HNN-extensions. We also prove that every Leibniz algebra with at most countable dimension embeds into a two-generator Leibniz algebra. [ABSTRACT FROM AUTHOR]

Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical representation numbers of G and give exact formulas for these numbers using multiplicities of polynomials that are defined by the Cayley–Menger determinant. One of the main results of the paper are explicit formulas for the representation numbers of the join of graphs which are obtained from W. Kuperberg's type theorem for two-distance sets. [ABSTRACT FROM AUTHOR]

Let J be a special Jordan algebra and let c n (J) be its corresponding codimension sequence. The aim of this paper is to prove that in case J is finite dimensional, such a sequence is polynomially bounded if and only if the variety generated by J does not contain U J 2 , the special Jordan algebra of 2 × 2 upper triangular matrices. As an immediate consequence, we prove that U J 2 is the only finite dimensional special Jordan algebra that generates a variety of almost polynomial growth. [ABSTRACT FROM AUTHOR]

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are γ -positive polynomials and can be interpreted as h -polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to s -inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain s -inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q. In this paper, for q even, we give a simple way to construct small degree polynomials over F q with no roots in the set μ q + 1 = { a ∈ F q 2 | a q + 1 = 1 } of the (q + 1) -th roots of unity. Further, we use these small degree polynomials over F q to construct and classify certain new families of permutation quadrinomials over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study some arithmetic properties of certain Toeplitz-type matrices over finite fields. For example, suppose { α 1 , ... , α n } are all the nonzero k -th powers in a finite field and for 1 ⩽ i , j ⩽ n , we set x i j = { 1 α i − α j if i ≠ j , 0 if i = j. We give an explicit formula for the characteristic polynomial of [ x i j ] 1 ⩽ i , j ⩽ n and confirm several conjectures posed by Zhi-Wei Sun. [ABSTRACT FROM AUTHOR]

Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics. In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, we provide a new construction of MOLS on size a prime power. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of q elements with characteristic p. Let Z q denote the unramified extension of the p -adic integers Z p with residue field F q. In this paper, we investigate the q -divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Z q / p m Z q , where the n variables of the polynomials are restricted to run through a box lifting F q n. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q -divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field F q (the case m = 1). Taking q = p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case. [ABSTRACT FROM AUTHOR]

Q -Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q -Gorenstein toric contact manifold are related to • the Ehrhart (quasi-)polynomial of its toric diagram; • the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity; • the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization. [ABSTRACT FROM AUTHOR]

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ -Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role. [ABSTRACT FROM AUTHOR]

Recently, Hou, Mu and Zeilberger introduced a new process of polynomial reduction for hypergeometric terms, which can be used to prove and generate hypergeometric identities automatically. In this paper, we extend this polynomial reduction to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of π -series involving Domb numbers and Franel numbers, and new families of congruences for Franel numbers and Delannoy numbers. [ABSTRACT FROM AUTHOR]

Recently Marcus, Spielman and Srivastava proved Weaver's KS r conjecture, which gives a positive solution to the Kadison-Singer problem. In [13,10] , Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KS r conjecture. In this paper, we present a new bound in Weaver's KS r conjecture for the arbitrary-rank case. To do that, we introduce the definition of (k , m) -characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle bound when r = 2 [8] and with the Ravichandran-Leake bound when r > 2 [21]. For the higher-rank case, we sharpen the previous bounds from Cohen and Brändén. [ABSTRACT FROM AUTHOR]

Introduced by Benhamouda and Lin [TCC'20], a multi-party reusable non-interactive secure computation protocol (mrNISC) consists of a commitment phase and an unbounded number of computation phases. In the commitment phase, a number of parties first commit to their input in a single broadcast round. Later in a computation phase, any subset of the parties can compute a function on their joint input by each sending a single broadcast message. Benhamouda and Lin [TCC'20] constructed the first mrNISC for all functions based on standard hardness assumptions in pairing groups. Soon after their work, two concurrent papers by Benhamouda et al. [EUROCRYPT'21] and Ananth et al. [EUROCRYPT'21] constructed mrNISC for all functions based on the hardness of LWE with super-polynomial modulus-to-noise ratio. In this work we build the first mrNISC for all functions based solely on LWE with polynomial modulus-to-noise ratio. We thus place mrNISC in the same category as public-key encryption and leveled fully homomorphic encryption in terms of the required LWE hardness assumption. We achieve our result by carefully introducing a bootstrapping step in the construction of Benhamouda et al. [ABSTRACT FROM AUTHOR]

In this paper we provide a characterization of the complexity class PSPACE by using a purely continuous model defined with polynomial ordinary differential equations. [ABSTRACT FROM AUTHOR]

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions on n for the unsigned and signed generalized Lucas polynomials f n (x) and g n (x) being permutations over the prime field. Moreover, the permutation behaviours of these polynomials over non-prime fields are also discussed. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low c -differential uniformity and boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over F 2 n , we determine the c -differential uniformity and boomerang uniformity of these permutation polynomials: (1) f 1 (x) = x + Tr 1 n (x 2 k + 1 + 1 + x 3 + x + u x) , where n = 2 k + 1 , u ∈ F 2 n with Tr 1 n (u) = 1 ; (2) f 2 (x) = x + Tr 1 n (x 2 k + 3 + (x + 1) 2 k + 3) , where n = 2 k + 1 ; (3) f 3 (x) = x − 1 + Tr 1 n ((x − 1 + 1) d + x − d) , where n is even and d is a positive integer. The results show that the involutions f 1 (x) and f 2 (x) are APcN functions for c ∈ F 2 n ﹨ { 0 , 1 }. Moreover, the boomerang uniformity of f 1 (x) and f 2 (x) can attain 2 n. Furthermore, we generalize some previous works and derive the upper bounds on the c -differential uniformity and boomerang uniformity of f 3 (x). [ABSTRACT FROM AUTHOR]

Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two "tuples of tuples" of indeterminates. We construct a family of linear bases of the fermionic Fock space that are indexed by such data and prove that they correspond to the multi-Schur functions through the boson-fermion correspondence. By focusing on some special bases, which we call refined bases, we give a straightforward method of expanding a multi-Schur function in the refined dual Grothendieck polynomials. We also present a sufficient condition for a multi-Schur function to have its Hall-dual function in the completed ring of symmetric functions. [ABSTRACT FROM AUTHOR]

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over F 2 m in Zieve's paper [30] . We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over F 2 m . Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials. [ABSTRACT FROM AUTHOR]

In this paper, the propagation of the Lamb waves in the GaAs-FGPM-AlAs sandwich plate is studied. Based on the orthogonal function, Legendre polynomial series expansion is applied along the thickness direction to obtain the Lamb dispersion curves. The convergence and accuracy of this polynomial method are discussed. In addition, the influences of the volume fraction p and thickness h F G P M of the FGPM middle layer on the Lamb dispersion curves are developed. The numerical results also show differences between the characteristics of Lamb dispersion curves in the sandwich plate for various gradient coefficients of the FGPM middle layer. In fact, if the volume fraction p increases the phase velocity will increases and the number of modes will decreases at a given frequency range. All the developments performed in this paper were implemented in Matlab software. The corresponding results presented in this work may have important applications in several industry areas and developing novel acoustic devices such as sensors, electromechanical transducers, actuators and filters. [ABSTRACT FROM AUTHOR]

This paper presents a variational reconstruction for the high order finite volume method in solving the two-dimensional Navier–Stokes equations on arbitrary unstructured grids. In the variational reconstruction, an interfacial jump integration is defined to measure the jumps of the reconstruction polynomial and its spatial derivatives on each cell interface. The system of linear equations to determine the reconstruction polynomials is derived by minimizing the total interfacial jump integration in the computational domain using the variational method. On each control volume, the derived equations are implicit relations between the coefficients of the reconstruction polynomials defined on a compact stencil involving only the current cell and its direct face-neighbors. The reconstruction and time integration coupled iteration method proposed in our previous paper is used to achieve high computational efficiency. A problem-independent shock detector and the WBAP limiter are used to suppress non-physical oscillations in the simulation of flow with discontinuities. The advantages of the finite volume method using the variational reconstruction over the compact least-squares finite volume method proposed in our previous papers are higher accuracy, higher computational efficiency, more flexible boundary treatment and non-singularity of the reconstruction matrix. A number of numerical test cases are solved to verify the accuracy, efficiency and shock-capturing capability of the finite volume method using the variational reconstruction. [ABSTRACT FROM AUTHOR]

Andô proved a dilation result that implies his celebrated inequality which says that if T 1 and T 2 are commuting contractions on a Hilbert space, then for any polynomial p in two variables, ‖ p ( T 1 , T 2 ) ‖ ≤ ‖ p ‖ D 2 , where D 2 is the bidisk in C 2 . The main goal of the present paper is to find analogues of Andô's results for the elements of the bi-ball P n 1 , n 2 which consists of all pairs ( X , Y ) of row contractions X : = ( X 1 , … , X n 1 ) and Y : = ( Y 1 , … , Y n 2 ) which commute, i.e. each entry of X commutes with each entry of Y . The results are obtained in a more general setting, namely, when X and Y belong to noncommutative varieties V 1 and V 2 determined by row contractions subject to constraints such as q ( X 1 , … , X n 1 ) = 0 and r ( Y 1 , … , Y n 2 ) = 0 , q ∈ P , r ∈ R , respectively, where P and R are sets of noncommutative polynomials. We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Andô's dilation theorem for commuting contractions, Sz.-Nagy–Foiaş commutant lifting theorem, and Schur's representation for the unit ball of H ∞ , in the framework of noncommutative varieties and Poisson kernels on Fock spaces. This leads to one of the main results of the paper, an Andô type inequality on noncommutative varieties, which, in the particular case when n 1 = n 2 = 1 and T 1 and T 2 are commuting contractive matrices with spectrum in the open unit disk D : = { z ∈ C : | z | < 1 } , takes the form ‖ p ( T 1 , T 2 ) ‖ ≤ min ⁡ { ‖ p ( B 1 ⊗ I C d 1 , φ 1 ( B 1 ) ) ‖ , ‖ p ( φ 2 ( B 2 ) , B 2 ⊗ I C d 2 ) ‖ } , where ( B 1 ⊗ I C d 1 , φ 1 ( B 1 ) ) and ( φ 2 ( B 2 ) , B 2 ⊗ I C d 2 ) are analytic dilations of ( T 1 , T 2 ) while B 1 and B 2 are the universal models associated with T 1 and T 2 , respectively. In this setting, the inequality is sharper than Andô's inequality and Agler–McCarthy's inequality. We obtain more general inequalities for arbitrary commuting contractive matrices and improve Andô's inequality for commuting contractions when at least one of them is of class C 0 . We prove that there is a universal model ( S ⊗ I ℓ 2 , φ ( S ) ) , where S is the unilateral shift and φ ( S ) is an isometric analytic Toeplitz operator on H 2 ( D ) ⊗ ℓ 2 , such that ‖ [ p r s ( T 1 , T 2 ) ] k ‖ ≤ ‖ [ p r s ( S ⊗ I ℓ 2 , φ ( S ) ) ] k ‖ , for any commuting contractions T 1 and T 2 on Hilbert spaces, any k × k matrix [ p r s ] k of polynomials in C [ z , w ] , and any k ∈ N . Analogues of this result for the bi-ball P n 1 , n 2 and for a class of noncommutative varieties are also considered. [ABSTRACT FROM AUTHOR]

The Hasse–Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse–Weil bound for the curve defined by y p − y = f ( x ) over the finite field F q , where p is the characteristic of F q . In 1993, Moreno–Moreno [7] gave an improvement to the Hasse–Weil bound for this family of curves. Recently, Kaufman and Lovett [4, FOCS2011] showed that the Hasse–Weil bound can be improved for this family of curves with f ( x ) = g ( x ) + h ( x ) , where g ( x ) is a polynomial of degree ≪ q and h ( x ) is a sparse polynomial of arbitrary degree but bounded weight degree. The other recent improvement by Rojas-Leon and Wan [9, Math. Ann. 2011] shows that an extra p can be removed for this family of curves if p is very large compared with polynomial degree of f ( x ) and log p ⁡ q . In this paper, we focus on the most interesting case for applications, namely p = 2 . We show that the Hasse–Weil bound for this special family of curves can be improved if q = 2 n with odd n ⩾ 3 which is the same case where Serre [10] improved the Hasse–Weil bound. However, our improvement is greater than Serre's and Moreno–Morenao's improvements for this special family of curves. Furthermore, our improvement works for p = 2 compared with the requirement of large p by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse–Witt invariant 0 and we show that the Hasse–Weil bound can be improved for any curves with the Hasse–Witt invariant 0. The main tool used in our proof involves Newton polygon and some results in algebraic geometry. [ABSTRACT FROM AUTHOR]

The classification of Curtis–Tits amalgams with connected, triangle free, simply-laced diagram over a field of size at least 4 was completed in [3] . Orientable amalgams are those arising from applying the Curtis–Tits theorem to groups of Kac–Moody type, and indeed, their universal completions are central extensions of those groups of Kac–Moody type. The paper [2] exhibits concrete (matrix) groups as completions for all Curtis–Tits amalgams with diagram A ˜ n − 1 . For non-orientable amalgams these groups are symmetry groups of certain unitary forms over a ring of skew Laurent polynomials. In the present paper we generalize this to all amalgams arising from the classification above and, under some additional conditions, exhibit their universal completions as central extensions of twisted groups of Kac–Moody type. [ABSTRACT FROM AUTHOR]

If one restricts an irreducible representation V λ of GL 2 n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ are even (resp. the conjugate partition λ ′ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q , t -generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q = 0 limit (Hall–Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using p -adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain p -adic measure counts. This approach provides a p -adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our p -adic method also leads to a generalized integral identity in terms of Littlewood–Richardson coefficients and Hall polynomials. [ABSTRACT FROM AUTHOR]

We identify a canonical parameter in the Chinese Remainder Theorem (CRT) and call it the “Reconstruction Coefficient”, (denoted by “ R C ”); and introduce the notions of “Partial” and “Full” Reconstruction. If the R C can be determined efficiently, then arithmetic operations that are (relatively) harder to realize in RNS; including Sign Detection, Base change/extension and Scaling or division by a constant can also be implemented efficiently. This paper therefore focuses on and presents two distinct methods to efficiently evaluate the R C at long wordlengths. A straightforward application of these methods leads to ultra-fast sign-detection. An independent contribution of this paper is to illustrate non-trivial trade-offs between run-time computation vs. pre-computation and look-up. We show a simple method to select the moduli which leads to both the (i) number of RNS channels n ; as well as (ii) the largest channel modulus m n satisfying { O ( n ) , O ( m n ) } ⪅ N ≡ the full-precision bit-length. The net result is that for many canonical operations; exhaustive look-up covering all possible input values is feasible even at long cryptographic bit-lengths N . Under fairly general and realistic assumptions about the capabilities of current hardware, the memory needed for exhaustive look-up tables is shown to be bounded by a low degree polynomial of n . Moreover, both methods to compute R C can achieve a delay of O ( lg n ) in a RN system with n channels. To the best of our knowledge, no other method published to date has shown a path to achieve that lower bound on the execution delay. Further, small values of channel moduli make it ideal to implement each individual RNS channel on a simple core in a many-core processor or as a distributed node, and our algorithms require a limited number of inter-channel communications, averaging O ( n ) . Results from a multi-core GPU implementation corroborate the theory. [ABSTRACT FROM AUTHOR]

Let f ∈ C [ − 1 , 1 ] and denote by E n ( f ) its degree of approximation by algebraic polynomials of degree < n . Assume that f changes its monotonicity, respectively, its convexity finitely many times, say s ≥ 2 times, in ( − 1 , 1 ) and we know that for q = 1 or q = 2 and some 1 < α ≤ 2 , such that q α ≠ 4 , we have E n ( f ) ≤ n − q α , n ≥ s + q + 1 . The purpose of this paper is to prove that the degree of comonotone, respectively, coconvex approximation, of f , by algebraic polynomials of degree < n , n ≥ N , is also ≤ c ( α , s ) n − q α , where the constant N depends only on the location of the extrema, respectively, inflection points in ( − 1 , 1 ) and on α . This answers, affirmatively, questions left open by the authors in papers with Kopotun (in Ukrainian Math. J.) and with Vlasiuk (see the list of references). [ABSTRACT FROM AUTHOR]

Let R be an integral domain. In this paper, we study the kernels of derivations of R [ 2 ] and closed polynomials in R [ n ] . Let D be an R -derivation of R [ 2 ] having divergence zero. In the first half of this paper, when R is a Q -domain, we give a necessary and sufficient condition for Ker D to be generated by one element over R . In the second half, we give some sufficient conditions for a non-constant polynomial to be closed in R [ n ] . [ABSTRACT FROM AUTHOR]

This is a sequel to the paper [4] , in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F q [ T ] -modules of rank r ≥ 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P -isogenies are reduced modulo a prime P . We also correct an error in [4] . [ABSTRACT FROM AUTHOR]

Let p n denote the orthonormal polynomials associated with a measure μ with compact support on the real line. In a recent paper, we showed there is a close relationship between the spacing of zeros of successive orthogonal polynomials p n , p n − 1 , and uniform bounds on the orthogonal polynomials in subintervals of the support. In this paper, we show there is also a relationship between asymptotics for the spacing of zeros of p n − 1 , p n , and pointwise asymptotics for the orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study bifurcations of small-amplitude limit cycles of Liénard systems of the form x ˙ = y − F (x) , y ˙ = − g (x) , where g (x) is a cubic polynomial, and F (x) is a smooth or piecewise smooth polynomial of degree n. By using involutions, we obtain sharp upper bounds of the number of small-amplitude limit cycles produced around a singular point for some systems of this type. [ABSTRACT FROM AUTHOR]

Abstract The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max ⁡ , +) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent. [ABSTRACT FROM AUTHOR]

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given F q -isogeny class. This work completes partial classification given in [9]. [ABSTRACT FROM AUTHOR]

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart polynomial are positive. Answering an Ehrhart positivity question posed on MathOverflow, Stanley provided an example of a non-Ehrhart-positive order polytope of dimension 21. Stanley's example comes from a certain family of order polytopes. In this paper, we study the Ehrhart positivity question on this family of polytopes. By giving explicit formulas for the coefficients of the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we determine the sign of each Ehrhart coefficient of each polytope in the family. As a consequence of our result, we conclude that for any positive integer d ≥ 21 , there exists an order polytope of dimension d that is not Ehrhart positive, and for any positive integer ℓ , there exists an order polytope whose Ehrhart polynomial has precisely ℓ negative coefficients, which answers a question posed by Hibi. We finish this article by discussing the existence of lower-dimensional order polytopes whose Ehrhart polynomials have a negative coefficient. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is twofold. Firstly, we generalize the notion of characteristic polynomials of hyperplane and toric arrangements to those of certain abelian Lie group arrangements. Secondly, we give two interpretations for the chromatic quasi-polynomials and their constituents through subspace and toric viewpoints. [ABSTRACT FROM AUTHOR]

In this paper, Runge–Kutta–Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge–Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type. We present SRK methods composed of L ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree L. Internal stability is maintained at large stage number through an ordering algorithm which limits internal amplification factors to 10 L 2. Test results for mildly stiff nonlinear advection-diffusion-reaction problems with moderate (≲1) mesh Péclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order. • We introduce the class of Runge-Kutta-Gegenbauer stability polynomials. • High order Factorized Runge-Kutta-Gegenbauer explicit methods are described. • Methods are shown to be well suited to mildly stiff advection-diffusion problems. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to develop foundations of umbral calculus on the space D ′ of distributions on R d , which leads to a general theory of Sheffer polynomial sequences on D ′. We define a sequence of monic polynomials on D ′ , a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D ′ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of binomial type on D ′ , and a lifting of a Sheffer sequence on R to a Sheffer sequence on D ′. Examples of lifted polynomial sequences include the falling and rising factorials on D ′ , Abel, Hermite, Charlier, and Laguerre polynomials on D ′. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role. [ABSTRACT FROM AUTHOR]

Abstract Recently, Nunge studied Eulerian polynomials on segmented permutations, namely generalized Eulerian polynomials , and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunge's conjecture. Our proof is based on Brändén's stable multivariate Eulerian polynomials. By acting on Brändén's polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunge's conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the factorizations of monic, bounded motion polynomials. We prove existence of factorizations, possibly after multiplication with a real polynomial Q , and provide an algorithm for computing Q and corresponding factorizations. The algorithm gives a much lower bound on the degree of the polynomial factor than the mere existence theorem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, several classes of complete permutation polynomials with the form (x p m − x + δ) s + a x p m + b x over F p n are proposed by the AGW criterion and determining the number of solutions of some equations. Our results also enrich constructions of known permutation polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct several new classes of complete permutation monomials a − 1 x d over a finite field F q n with exponents d = q n − 1 q − 1 + 1 , q p − 1 − 1 q − 1 + 1 , and q q − 1 − 1 q − 1 + 1 , respectively, where q = p k is a power of a prime number p. Our approach uses the AGW criterion (the multiplicative case) together with Dickson permutation polynomials and a class of exceptional polynomials respectively. One of our results confirms Conjecture 4.18 by G. Wu, N. Li, T. Helleseth, Y. Zhang in [42] under the assumption that the characteristic p is primitive modulo a prime number n + 1. Moreover, we show that Conjecture 4.18 is false in general using our approach and a counterexample is provided. We also re-confirm Conjecture 4.20 in [42] that was proved recently in [24] , and extend some of these recent results to more general n 's and more general a 's. [ABSTRACT FROM AUTHOR]

Abstract Subspace codes and particularly cyclic subspace codes have attracted a lot of research due to their applications in error correction for random network coding. In this paper, we explore the parameters of a class of the cyclic subspace codes whose subspace polynomials are more generalized. Therefore, some corollaries that extend the previously known results are obtained. Finally, a construction of k -dimensional cyclic subspace codes of size q N − 1 q − 1 and minimal distance 2 k − 2 is provided. [ABSTRACT FROM AUTHOR]

A new (k , n) threshold secret image sharing scheme with two decoding options is introduced in this paper. The proposed scheme combines color-black-and-white visual cryptography scheme (CBW-VCS) and polynomial based secret image sharing (PSIS) together, to offer stacking-to-see decryption and lossless image reconstruction. To construct the color shares, a general (k , n) threshold CBW-VCS is firstly given. A grayscale secret image is converted to a p -radix image and a binary image. The p -radix image is encrypted by the (k , n) PSIS under mod p operation, and n p -radix shadows are obtained. Then, a color share generation algorithm with data embedding is proposed to construct n color shares. Theoretical analysis and sufficient experiments are provided to show the effectiveness and advantages of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q , t) -Catalan numbers along with their corresponding γ -expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (− 1) -phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their q -analogues utilizing the newly obtained q - γ -expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely (2413 , 3142) and (1342 , 2431) , of length four, and consider such (− 1) -phenomenon for these two subsets as well. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first determine the minimum possible size of an F q -linear set of rank k in PG (1 , q n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F q -linear set of rank k in PG (2 , q n). In the case k = n , this confirms a conjecture by Sziklai in [9]. [ABSTRACT FROM AUTHOR]

Abstract The paper is concerned with the lowest vibration modes of multi-component rods and cylinders with alternating high contrast material properties of the components. It is demonstrated that these modes correspond to almost rigid body motions of the "stiffer" components. A general perturbation scheme is developed. At leading order polynomial frequency equations are derived, along with linear algebraic equations for the associated eigenforms. In addition, for multi-component rods, higher order corrections are also obtained. Examples of three- and five-component structures are presented, illustrating the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds. [ABSTRACT FROM AUTHOR]

Abstract Let F (z) be a Hecke–Maass form for S L (3 , Z) (or S L (2 , Z)), A F (n , 1) (or A F (n)) be its Fourier coefficients, and L (s , F) the Godement–Jacquet L -function attached to F. And let μ F (n) be the coefficients of the reciprocal of L (s , F) in the absolutely convergent region ℜ s > 1. In this paper we prove that μ F (n) is strongly orthogonal to 1-step nilsequence. As applications, we prove that the Möbius function μ (n) is strongly orthogonal to A F (n , 1) e (n α) and A F (n) 2 e (n α). More precisely, we establish ∑ n ≤ N μ (n) A F (n , 1) e (n α) ≪ c N exp ⁡ (− c log ⁡ N) , and ∑ n ≤ N μ (n) A F (n) 2 e (n α) ≪ A N (log ⁡ N) − A. Moreover, we also show that μ F (n) is asymptotically orthogonal to any A C 0 (d) function, and all MMA and MMLS flows. [ABSTRACT FROM AUTHOR]

Abstract Starting from our previous papers Angelini et al. (2018c) and Angelini et al. (2018a) , we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2 , 3 , 3 , 3 and rank 6, which are not identifiable over C but are identifiable over R. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion (Chiantini et al., 2014), we prove identifiability over C for polynomial vectors in many cases of sub-generic rank. [ABSTRACT FROM AUTHOR]