Related papers: Class invariants for certain non-holomorphic modul…
We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition…
We introduce special classes of irreducible representations of groups: thick representations and dense representations. Denseness implies thickness, and thickness implies irreducibility. We show that absolute thickness and absolute…
A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behaviour with…
Let $\Omega \subset \mathbb{C}^m$ be an open, connected and bounded set and $\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on $\Omega$. In this article we study quotient Hilbert modules obtained from submodules,…
We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…
The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…
Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given.…
In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $\mathbb R^n$, and having only finitely many zeros in this set. Following C. Bivi\`{a}-Ausina, we…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…
Simple function classes have emerged as toy problems to better understand in-context-learning in transformer-based architectures used for large language models. But previously proposed simple function classes like linear regression or…
We set up a formalism of Maurer-Cartan moduli sets for L-infinity algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other…
We give a number new examples analytically and numerically to confirm the Kohler conjecture. It turned out that for a rather large class of nonnegative functions the equality (A) hold.
A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the…
By viewing non-commutative polynomials, that is, elements in free associative algebras, in terms of linear representations, we generalize Horner's rule to the non-commutative (multivariate) setting. We introduce the concept of Horner…
It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural "derived" moduli spaces which are always smooth in an…
We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise…
We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite…
Let $\lambda$ be a general length function for modules over a Noetherian ring R. We use $\lambda$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$\lambda$. We show that the leading term $\mu$ of…