Equivariant Polynomial Research Articles

Hermite equivalence of polynomials

We compare results of Birch and Merriman (1972), Győry (1973, 1974) and Evertse and Győry (1991, 2017), which imply in a precise form that polynomials in $\mathbb {Z} [X]$ of given discriminant lie in finitely many ${\rm GL}_2(\mathbb {Z} )$-equivalen

Cocharacters for the weak polynomial identities of the Lie algebra of 3 × 3 skew-symmetric matrices

Let so3(K) be the Lie algebra of 3×3 skew-symmetric matrices over a field K of characteristic 0. The ideal I(M3(K),so3(K)) of the weak polynomial identities of the pair (M3(K),so3(K)) consists of the elements f(x1,…,xn) of the free associative algebra K〈X〉 with the property that f(a1,…,an)=0 in the algebra M3(K) of all 3×3 matrices for all a1,…,an∈so3(K). The generators of I(M3(K),so3(K)) were found by Razmyslov in the 1980s. In this paper the cocharacter sequence of I(M3(K),so3(K)) is computed. In other words, the GLp(K)-module structure of the algebra generated by p generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of SO3(K)-equivariant polynomial maps from the space of p-tuples of 3×3 skew-symmetric matrices into M3(K) (endowed with the conjugation action). In the special case p=3 the latter algebra is a module over a 6-variable polynomial subring in the algebra of SO3(K)-invariants of triples of 3×3 skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.

An Extension of Gröbner Basis Theory to Indexed Polynomials Without Eliminations

In computer algebra, it remains to be challenging to establish general computational theories for determining the equivalence of indexed polynomials. In previous work, the author solved the equivalence determination problem for Riemann tensor polynomials by extending Grobner basis theory. This paper extends the previous work to more general indexed polynomials that involve no eliminations of indices and functions, by the method of ST-restricted rings. A decomposed form of the Grobner basis of the defining syzygy set in each ST-restricted ring is provided, and then the canonical form of an indexed polynomial proves to be the normal form with respect to the Grobner basis in the ST-fundamental restricted ring.

Limit Cycles for a Class of Zp-Equivariant Differential Systems

In this paper, we study a class of [Formula: see text]-equivariant planar polynomial differential systems [Formula: see text]. It is shown that for any [Formula: see text] there is a differential system of the above type having at least [Formula: see text] limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

Distance invariant method for normalization of indexed differentials

A distance from free to dummy indices is defined. The distance is invariant with respect to both monoterm symmetries and bottom antisymmetry. Using the distance invariant, we present an index-replacement algorithm. We then develop two normalization algorithms. One is with respect to monoterm symmetries and has complexity lower than known algorithms; the other allows one to determine the equivalence of indexed polynomials in the Einstein summation ring R2[∂̸].

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n×n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2×2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of “chain structured tropical matrix semigroups”; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3×3 upper triangular tropical matrices.

Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Abstract Let a ⁢ ( x ) {a(x)} be non-constant and let b j ⁢ ( x ) {b_{j}(x)} , for j = 0 , 1 , 2 , 3 {j=0,1,2,3} , be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation a ⁢ ( x ) ⁢ y ˙ = b 1 ⁢ ( x ) ⁢ y + b 3 ⁢ ( x ) ⁢ y 3 {a(x)\dot{y}=b_{1}(x)y+b_{3}(x)y^{3}} , with b 3 ⁢ ( x ) ≠ 0 {b_{3}(x)\neq 0} , and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a ⁢ ( x ) ⁢ y ⁢ y ˙ = b 0 ⁢ ( x ) + b 2 ⁢ ( x ) ⁢ y 2 {a(x)y\dot{y}=b_{0}(x)+b_{2}(x)y^{2}} , with b 2 ⁢ ( x ) ≠ 0 {b_{2}(x)\neq 0} , have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

Normalization in Riemann Tensor Polynomial Ring

It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Grobner basis theory is not yet powerful enough to deal with ideals that cannot be finitely generated. This paper solves the problem by extending Grobner basis theory. First, the polynomials are described via an infinitely generated free commutative monoid ring. The authors then provide a decomposed form of the Grobner basis of the defining syzygy set in each restricted ring. The canonical form proves to be the normal form with respect to the Grobner basis in the fundamental restricted ring, which allows one to determine the equivalence of polynomials. Finally, in order to simplify the computation of canonical form, the authors find the minimal restricted ring.

Relative equivariants under compact Lie groups

In this work we obtain the general form of polynomial mappings that commute with a linear action of a relative symmetry group. The aim is to give results for relative equivariant polynomials that correspond to the results for relative invariants obtained in a previous paper (Baptistelli and Manoel (2013) [5]). We present an algorithm to compute generators for relative equivariant submodules from the invariant theory applied to the subgroup formed only by the symmetries. The same method provides, as a particular case, generators for equivariants under the whole group from the knowledge of equivariant generators by a smaller subgroup, which is normal of finite index.

Symmetries of a Flat Cosymbol Algebra of Differential Operators

In this paper, a structure theorem for the symmetries of a graded flat cosymbol algebra of differential operators is proved. Together with a lemma on equivariant polynomials also proved in the paper, this theorem gives an upper bound on the dimension of the graded Lie algebra associated with the symmetries of geodesic flow on a smooth variety. Bibliography: 14 titles

Symmetric Self-Adjoint Hopf Categories and a Categorical Heisenberg Double

We define what we call a symmetric self-adjoint Hopf structure on a semisimple abelian category, which is an analog of Zelevinsky’s positive self-adjoint Hopf algebra structure for categories. As examples we exhibit this structure on the categories of polynomial functors and equivariant polynomial functors and obtain a categorical manifestation of Zelevinsky’s decomposition theorem involving them. It follows from the work of Zelevinsky that every positive self-adjoint Hopf algebra A admits a Fock space action of the Heisenberg double (A,A). We show that the notion of symmetric self-adjoint Hopf category leads naturally to the definition of a categorical analog of such an action and that every symmetric self-adjoint Hopf category admits such an action.

Resultant of an equivariant polynomial system with respect to the symmetric group

Given a system of n⩾2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols, it is proved that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case

Let f=(f1,…,fm) and g=(g1,…,gm) be two sets of m≥1 nonlinear polynomials in K[x1,…,xn] (K being a field). We consider the computational problem of finding–if any–an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isomorphism of Polynomials with one Secret (IP1S) and is a fundamental problem in multivariate cryptography. Amongst its applications, we can cite Graph Isomorphism (GI) which reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables, according to Agrawal and Saxena. The main result is a randomized polynomial-time algorithm for solving IP1S for quadratic instances–a particular case of importance in cryptography.To this end, we show that IP1S for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory. We show that we can essentially linearize the problem by reducing quadratic-IP1S to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski (ISSAC 1997) to be equivalent to finding an invertible matrix in the linear space Kn×n of n×n matrices over K and to compute the square root in a certain representation in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing a representation of the square root of a matrix in Kn×n, for various fields (including finite fields), as a product of two matrices. Each coefficient of these matrices lies in an extension field of K of polynomial degree. We then consider #IP1S, the counting version of IP1S for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general Isomorphism of Polynomials (IP) problem where we allow an invertible linear transformation on the variables and on the set of polynomials. A randomized polynomial-time algorithm for solving IP when f=(x1d,…,xnd) is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (i.e. a matrix whose components are linear polynomials). This extends to IP a result of Kayal obtained for PolyProj.

Spectral equivalence of matrix polynomials and the Index Sum Theorem

The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and ℓ-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. The properties of ℓ-ifications are studied in depth; in this process a fundamental result on matrix polynomials, the “Index Sum Theorem”, is recovered and extended to arbitrary fields. Although this result is known in the systems theory literature for real matrix polynomials, it has remained unnoticed by many researchers. It establishes that the sum of the (finite and infinite) partial multiplicities, together with the (left and right) minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial. The “Index Sum Theorem” turns out to be a key tool for obtaining a number of significant results: on the possible sizes and degrees of ℓ-ifications and companion forms, on the minimal index preservation properties of companion forms of arbitrary degree, as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree. This paper presents many new results, blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields. Therefore we have written the paper in an expository and self-contained style that makes it accessible to a wide variety of readers.

The discrete Markus–Yamabe problem for symmetric planar polynomial maps

We probe deeper into the discrete Markus–Yamabe question for polynomial planar maps and into the normal form for those maps which answer this question in the affirmative. Furthermore, in a symmetric context, we show that the only nonlinear equivariant polynomial maps providing an affirmative answer to the discrete Markus–Yamabe question are those possessing Z2 as their group of symmetries. We use this to establish two new tools which give information about the spectrum of a planar polynomial map.

Compression of finite group actions and covariant dimension, II

Let G be a finite group and φ : V → W an equivariant polynomial map between finite dimensional G-modules. We say that φ is faithful if G acts faithfully on φ ( V ) . The covariant dimension of G is the minimum of the dimension of φ ( V ) ¯ taken over all faithful φ. In [Hanspeter Kraft, Gerald W. Schwarz, Compression of finite group actions and covariant dimension, J. Algebra 313 (1) (2007) 268–291] we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of the paper, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of our previous work.

Invariants, equivariants and characters in symmetric bifurcation theory

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions’. We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group O(3).

AUTOMORPHIC EQUIVALENCE PROBLEM FOR FREE ASSOCIATIVE ALGEBRAS OF RANK TWO

Let K 〈x,y〉 be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K 〈x,y〉 are equivalent under an automorphism of K 〈x,y〉. A modification of our algorithm solves the problem whether or not an element in K 〈x,y〉 is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K 〈x,y〉. An algorithm for equivalence of polynomials under automorphisms of ℂ[x,y] was presented by Wightwick. Another, much simpler algorithm for automorphic equivalence of two polynomials in K[x,y] for any algebraically closed constructive field K was given by Makar-Limanov, Shpilrain, and Yu. In our approach we combine an idea of the latter three authors with an idea from the unpublished thesis of Lane used to describe automorphisms which stabilize elements of K 〈x,y〉. This also allows us to give a simple proof of the corresponding result for K[x,y] obtained by Makar-Limanov, Shpilrain, and Yu.

Equivalence of polynomials under automorphisms of [formula omitted

Let K [ x , y ] be the algebra of polynomials in two variables over an arbitrary field K . We show that if the maximum of the x - and y -degrees of a given polynomial p ( x , y ) cannot be decreased by a single triangular or linear automorphism of K [ x , y ] , then it cannot be decreased by any automorphism of K [ x , y ] . If K is an algebraically closed constructible field, this result yields an algorithm for deciding whether or not two polynomials p , q ∈ K [ x , y ] are equivalent under an automorphism of K [ x , y ] . We also show that if there is an automorphism of K [ x , y ] taking p to q , then it is “almost” unique. More precisely: if an automorphism α of K [ x , y ] is not conjugate to a triangular or linear automorphism, then any polynomial invariant (or even semiinvariant) under α is a constant.

Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11

In this article, a systematic procedure has been explored to studying general Z q -equivariant planar polynomial Hamiltonian vector fields for the maximal number of closed orbits and the maximal number of limit cycles after perturbation. Following the procedure by taking special consideration of Z 12 -equivariant vector fields of degree 11, the maximal of 99 closed orbits are obtained under a well-defined coefficient group. Consequently, perturbation parameter control in limit cycle computation leads to the existence of 121 limit cycles in the perturbed Hamiltonian vector field, which gives rise to the lower bound of Hilbert number of 11th-order systems as H (11) ⩾ 11 2 . Two conjectures are proposed regarding the maximal number of closed orbits for equivariant polynomial Hamiltonian vector fields and the maximal number of limit cycles bifurcated from the well defined Hamiltonian vector fields after perturbation.