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This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n + 1 has at least n + [ n 4 ] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center. [ABSTRACT FROM AUTHOR]

The Gyárfás-Sumner conjecture says that for every forest H and every integer k , if G is H -free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H , if G is H -free and does not contain K t , t as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H , the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer d ≥ 2 , if G is H -free and does not contain as a subgraph the complete d -partite graph with parts of cardinality t , then its chromatic number is at most polynomial in t. [ABSTRACT FROM AUTHOR]

The maximum number of centers is an open problem proposed by Gasull for planar polynomial differential systems of degree n with n ≥ 4. In this paper we study the problem for planar polynomial Kolmogorov differential systems of degree n , prove that the maximum number of centers is exactly seven for planar quartic polynomial Kolmogorov differential systems, and give the upper and lower bound for the maximum number of centers that the Kolmogorov differential systems of degree n can have. [ABSTRACT FROM AUTHOR]

Let U q (g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g , and let V be the corresponding type 1 finite-dimensional irreducible representation of U q (g). Starting from this data, the R -matrix formalism for quantum groups outputs a Hopf algebra U R λ (g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of U q (g) by adding a single invertible quantum Cartan element. We simultaneously establish that U R λ (g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining U R λ (g) , with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End (V). [ABSTRACT FROM AUTHOR]

In this paper we classify all monic, quartic, polynomials d (x) ∈ Z [ x ] for which the Pell equation f (x) 2 − d (x) g (x) 2 = 1 has a non-trivial solution with f (x) , g (x) ∈ Z [ x ]. [ABSTRACT FROM AUTHOR]

In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for the intervals (0 , 1) and (− N , − N + 1) for N ≥ 5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials. [ABSTRACT FROM AUTHOR]

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε > 0 , if an n -vertex graph G contains ε n 2 edge-disjoint copies of H then G contains δ n v (H) copies of H for some δ = δ (ε , H) > 0. The current proofs of the removal lemma give only very weak bounds on δ (ε , H) , and it is also known that δ (ε , H) is not polynomial in ε unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ (ε , H) depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson on this topic. [ABSTRACT FROM AUTHOR]

In 1971, Wilson (1971) proved that "almost all" finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we present a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies. Additionally, we provide some applications to recent models of games on networks such that Patacchini-Zenou's model about juvenile delinquency and conformism (Patacchini and Zenou, 2012), Calvó-Armengol-Patacchini-Zenou's model about social networks in education (Calvó-Armengol et al., 2009), Konig-Liu-Zenou's model about R&D networks (König et al., 2019), Helsley-Zenou's model about social networks and interactions in cities (Helsley and Zenou, 2014). [ABSTRACT FROM AUTHOR]

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving systems acted by Z d to the case of measure-preserving systems acted by Z d with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for K -systems acted by Z d. [ABSTRACT FROM AUTHOR]

We study the category of representations of sl m + 2 n over a field of characteristic p with p ≫ 0 , whose p -character is a nilpotent whose Jordan type is the two-row partition (m + n , n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n = 1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger. [ABSTRACT FROM AUTHOR]

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation. [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime, and R = G R (p 2 , p 2 r) is a Galois ring of characteristic p 2 , whose cardinality is p 2 r. For any subset D ⊆ R , let N (D , n , b) and N f (D , n , b) , respectively, be the number of n -subsets, denoted by S , in D such that ∑ x ∈ S x = b and ∑ x ∈ S f (x) = b , where f (x) ∈ R [ x ] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N (D , n , b) and N f (D , n , b) under some certain constraints of D and f (x). The special cases D = { a k : a ∈ R ⁎ } and f (x) = x k were studied by the authors in a previous paper, where R ⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings. [ABSTRACT FROM AUTHOR]

In this paper, we give characterization of quadratic ε -canonical number system (ε −CNS) polynomials for all values ε ∈ [ 0 , 1). Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials (ε = 0) and quadratic SCNS polynomials (ε = 1 / 2). This result is a consequence of our new characterization results of ε -shift radix systems (ε −SRS) in the two-dimensional case and their relation to quadratic ε −CNS polynomials. [ABSTRACT FROM AUTHOR]

Let T m (N , 2 k) be the m th Hecke operator on the space S (N , 2 k) of cuspforms of weight 2 k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of T 2 (N , 2 k) does not vanish when 2 and N are coprime. [ABSTRACT FROM AUTHOR]

This paper devotes to the study of planar polynomial differential systems with homogeneous nonlinearities of degree n > 1. We are concerned with the maximum number of limit cycles surrounding the origin of such systems, denoted by H o (n). By means of the second order analysis using the theories of Melnikov functions, we provide new estimates for H o (n) restricted to the cases where the origin is a focus, a node, a saddle or a nilpotent singularity. In particular, H o (n) ≥ n for each n in the case of focus. To the best of our knowledge, this improves the previous works in the literature. [ABSTRACT FROM AUTHOR]

For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [ABSTRACT FROM AUTHOR]

Let F be an infinite field and M 1 , 1 (F) be the algebra of 2 × 2 matrices over F endowed with non-trivial Z 2 -grading. We consider the involutions ⁎ defined on M 1 , 1 (F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M 1 , 1 (F) , ⁎) and determine the generators of its ideal of (Z 2 , ⁎) -identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z 2 , ⁎) -polynomials, when the characteristic of F is different from 2. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation, ν : K → R ∪ { − ∞ } , with a more general class of hyperfield homomorphisms, H → T , which satisfy a relative algebraic closure condition. The map η : T C → T , η (x) : = log (| x |) , where T C is the tropical complex hyperfield, provides an example of such a hyperfield homomorphism. [ABSTRACT FROM AUTHOR]

Let K be a field and let J n , k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n − k over K. Then one can consider the decomposition J n , k = B n − k ⊕ D k , where B n − k represents the corresponding Jordan algebra, denoted as B n − k = K ⊕ V. In this algebra, the restriction of b on the (n − k) -dimensional subspace V is non-degenerate, while D k accounts for the degenerate part of J n , k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for J n , k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T -ideal of polynomial identities of J n , k. Over a field of characteristic zero, we also prove that the ideal of identities of J n , k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z 2 -grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace. [ABSTRACT FROM AUTHOR]

Let f (x 1 , x 2) be a two-variable polynomial over a finite field of characteristic p , and assume that its convex hull is a triangle Δ. The goal of this paper is to study the Newton polygon NP (f , χ) of the L -function associated to f and a finite character χ of Z p. It is a partial generalization of [DWX] , in which the authors studied polynomials of one variable. In this paper, we prove an improved lower bound IHP (Δ) for NP (f , χ) , which, when f is non-ordinary, is strictly higher than the classical Hodge polygon. Moreover, we prove that if NP (f , χ) and IHP (Δ) coincide at one certain point, then they coincide at infinitely many points. This recovers the key property of the proof from [DWX]. [ABSTRACT FROM AUTHOR]

In this paper we derive a closed formula for the (Z × Z 2) -graded dimension of the cyclotomic quiver Hecke superalgebra R Λ (β) associated to an arbitrary Cartan superdatum (A , P , Π , Π ∨) , polynomials (Q i , j (x 1 , x 2)) i , j ∈ I , β ∈ Q n + and Λ ∈ P +. As applications, we obtain a necessary and sufficient condition for which e (ν) ≠ 0 in R Λ (β). We construct an explicit monomial basis for the bi-weight space e (ν ˜) R Λ (β) e (ν ˜) , where ν ˜ is a certain specific n -tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α 1 + α 2 + ⋯ + α n with α 1 , ⋯ , α n distinct. We construct an explicit monomial basis of R Λ (β) and show that it is indecomposable in this case. [ABSTRACT FROM AUTHOR]

A constant term sequence is a sequence of rational numbers whose n -th term is the constant term of P n (x) Q (x) , where P (x) and Q (x) are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms. [ABSTRACT FROM AUTHOR]

In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions (⩾2) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the Makar-Limanov and Derksen invariant of generalised Asanuma varieties under certain hypotheses. We also establish a necessary and sufficient condition for certain generalised Asanuma varieties to be isomorphic to polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space. We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for "solving" positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots. Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight. [ABSTRACT FROM AUTHOR]

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as ϵ → 0. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called V I 3). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded. [ABSTRACT FROM AUTHOR]

A color correction method for balancing the color appearances, among a group of images, about a specified object or scene, such as panoramic images and object movies, is developed and tested in this paper. In order to increase the running speed of color correction and reduce the out-of-gamut color pixels, we introduce the selection of principal regions. The average color values of principal regions are used to construct the low-degree (up to degree two) polynomial mapping functions from the source images to the corrected images. The functions are run in the decorrelated color spaces. Our method is tested using real and synthetic images. The results of these tests show the proposed method can get a better performance than other existing methods. [Copyright &y& Elsevier]

This paper presents an E -derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K - E -derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables. [ABSTRACT FROM AUTHOR]

It is shown in the paper that each Hecke symmetry R with the R -symmetric algebra freely generated by 3 commuting elements is determined by a bivector and a symmetric bilinear form on a 3-dimensional vector space. A general formula for such Hecke symmetries is given and the equivalence classes are described. [ABSTRACT FROM AUTHOR]

This paper is concerned with Smith forms of bivariate polynomial matrices. For a bivariate polynomial square matrix with the determinant being the product of two distinct and irreducible univariate polynomials, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms. [ABSTRACT FROM AUTHOR]

For any positive integer t , a t-broom is a graph obtained from K 1 , t + 1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t -brooms, we have χ (G) = o (ω (G) t + 1) , where χ (G) and ω (G) are the chromatic number and clique number of G , respectively. When t = 2 , this answers a question of Schiermeyer and Randerath. Moreover, for t = 2 , we strengthen the bound on χ (G) to 7 ω (G) 2 , confirming a conjecture of Sivaraman. For t ≥ 3 and { t -broom, K t , t }-free graphs, we improve the bound to o (ω t). [ABSTRACT FROM AUTHOR]

In this paper, we study the semilinear elliptic equation − Δ u + a (x) | u | p − 2 u − b (x) | u | q − 2 u = 0 on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N ≥ 1 , where 2 ≤ p < q < + ∞. We first prove the existence of positive solutions to the above equation with constant coefficients a ¯ , b ¯. Then we establish a decomposition of Palais-Smale sequences for the functional with variable coefficients a (x) , b (x) , which tend to the constants a ¯ , b ¯ at infinity. [ABSTRACT FROM AUTHOR]

In the present paper we study U T (D 1 , ... , D n) , a G -graded algebra of block triangular matrices where G is a group and the diagonal blocks D 1 , ... , D n are graded division algebras. We prove that any two such algebras are G -isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of U T n to the case in which the diagonal blocks D 1 , ... , D n are all isomorphic. [ABSTRACT FROM AUTHOR]

Let I = 〈 f 1 , ... , f m 〉 ⊂ Q [ x 1 , ... , x n ] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are given approximations { z 1 , ... , z k } ⊂ C n for the common roots { ξ 1 , ... , ξ k } = V (I) ⊆ C n. In this paper we show how to construct and certify the rational entries of Hermite matrices for I from the approximate roots { z 1 , ... , z k }. When I is non-radical, we give methods to construct and certify Hermite matrices for I from the approximate roots. Furthermore, we use signatures of these Hermite matrices to give rational certificates of non-negativity of a given polynomial over a (possibly positive dimensional) real variety, as well as certificates that there is a real root within an ε distance from a given point z ∈ Q n. [ABSTRACT FROM AUTHOR]

Considering a multivariate polynomial ideal over a given field as a vector space, we investigate for such an ideal a particular linear basis, a so-called staggered linear basis, which contains a Gröbner basis as well. In this paper, we present a simple and efficient algorithm to compute a staggered linear basis. The new framework is equipped with some novel criteria (including both Buchberger's criteria) to detect superfluous reductions. The proposed algorithm has been implemented in Maple, and we provide an illustrative example to show how it works. Finally, the efficiency of this algorithm compared to the existing methods is discussed via a set of benchmark polynomials. [ABSTRACT FROM AUTHOR]

Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations. [ABSTRACT FROM AUTHOR]

This paper deals with the fundamental period π ˜ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field F q (θ). Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in π ˜ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z 2 , the cyclic group of order 2. In this case A is said to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A = A 0 ⊕ A 1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra U T n (F) of n × n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities. [ABSTRACT FROM AUTHOR]

Let P k be the graded polynomial algebra F 2 [ x 1 , x 2 , ... , x k ] with the degree of each generator x i being 1, where F 2 denote the prime field with two elements. The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra P k as a module over the mod-2 Steenrod algebra A. Equivalently, we want to find a vector space basis for F 2 ⊗ A P k in each degree. In this paper, we study a generating set for the kernel of Kameko's squaring operation S q ˜ ⁎ 0 : F 2 ⊗ A P k ⟶ F 2 ⊗ A P k in a so-called generic degree. By using this result, we explicitly compute the hit problem for k = 5 in the respective generic degree. [ABSTRACT FROM AUTHOR]

As is well-known, the choice of variable ordering while computing cylindrical algebraic decomposition (CAD) has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called "the perfect elimination orderings", PEO), naturally preserve chordality, which is well compatible with an important (variable) sparsity pattern called "the correlative sparsity". If the associated graph of the polynomial system in question is chordal (resp. , is nearly chordal), then a PEO of the associated graph (resp. , of a minimal chordal completion of the associated graph) can be a better variable ordering for the CAD computation than other naive variable orderings in the sense that it results in a much smaller full set of projection polynomials and thus more efficient computation. A new (m , d) -property of the full set of CAD projection polynomials obtained via a PEO is given, which indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller "size". Furthermore, combining the lower-tree-height rule with Brown's heuristics, a new procedure is proposed to choose better PEOs for CAD computation. [ABSTRACT FROM AUTHOR]

Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as D -finite sequences. A subclass are C -finite sequences satisfying a linear recurrence with constant coefficients. We investigate the set of sequences which satisfy linear recurrence equations with coefficients that are C -finite sequences and call them C 2 -finite sequences. These sequences are a natural generalization of holonomic sequences. In this paper, we show that C 2 -finite sequences form a difference ring and provide methods to compute in this ring. Furthermore, we provide an analogous construction for D 2 -finite sequences, i.e., sequences satisfying a linear recurrence with holonomic coefficients. We show that these constructions can be iterated and obtain an increasing chain of difference rings. [ABSTRACT FROM AUTHOR]

In this paper, we present a q -analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the q -Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given q -hypergeometric term. The structure theorem enables us to generalize the q -polynomial reduction to the rational case, which can be used in the automatic proof and discovery of q -identities. As applications, several q -analogues of series for π are presented. [ABSTRACT FROM AUTHOR]

This is a paper about c -functions and Macdonald polynomials. There are c -function formulas for E -expansions of P λ and A λ + ρ , principal specializations of P λ and E μ , for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from [19]. In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element h and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity. [ABSTRACT FROM AUTHOR]

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6] , but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q = 0 and/or t = 0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials. [ABSTRACT FROM AUTHOR]

In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra D n of n × n diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of D n + 1 endowed with a degenerate trace, to those of D n with the corresponding trace. This allows us to determine the generators of the trace T-ideal of D 3. In the second part we study commutative subalgebras of M k (F) , denoted by C k of the type F + J that can be endowed with the so-called strange traces: tr (a + j) = α a + β j , for any a + j ∈ C k , α , β ∈ F. Here J is the radical of C k. In case β = 0 such a trace is degenerate, and we study the trace identities satisfied by the algebra C k , for every k ≥ 2. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth. In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in ([7]). [ABSTRACT FROM AUTHOR]

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials. [ABSTRACT FROM AUTHOR]

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f 1 , ... , f N) ∈ C [ x 1 , ... , x n ] N , we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α -theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria. [ABSTRACT FROM AUTHOR]

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables. [ABSTRACT FROM AUTHOR]