Necklaces,symmetries and self-reciprocal polynomials

The connection between a certain class of necklaces and self-reciprocal polynomials over finite fields is shown. For n?2, self-reciprocal polynomials of degree 2n arising from monic irreducible polynomials of degree n are shown to be either irreducible or the product or two irreducible factors which are necessarily reciprocal polynomials. Using DeBruijn's method we count the number of necklaces in this class and hence obtain a formula for the number of irreducible self-reciprocal polynomials showing that they exist for every even degree. Thus every extension of a finite field of even degree can be obtained by adjoining a root of an irreducible self-reciprocal polynomial.     

Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces

The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.     

Positivity of quiver coefficients through Thom polynomials

We use the Thom Polynomial theory developed by Fehér and Rimányi to prove the component formula for quiver varieties conjectured by Knutson, Miller, and Shimozono. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of Buch and Fulton are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in K-theoretic versions of the component formula.     

Heat polynomials and Lie point symmetries

The classical heat polynomials are polynomial solutions of the heat equation. We demonstrate the generation of such polynomials through the medium of the group theoretical properties of the equation. A generalised procedure for the generation of polynomial solutions is presented and this is extended to the construction of related polynomials.     

Oriented bordism of codimension one immersions of 7-manifolds and relative Thom polynomials

We give a formula to detect the oriented bordism class of a codimension one immersion of an oriented 7-manifold in terms of singularities of its singular Seifert surface, that is, a generic map from a compact 8-manifold which extends the given immersion. Our argument involves a study of a relative version of the Thom polynomials for certain singularities of generic maps from manifolds with non-empty boundaries.     

Direct and inverse theorems for Bernstein polynomials with inner singularities

We introduce a new type of Bernstein polynomials, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type of combinations are obtained.     

The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities

Journal of Algebraic Combinatorics - A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated...     

Mirror map for Fermat polynomials with a nonabelian group of symmetries

Theoretical and Mathematical Physics - We study Landau–Ginzburg orbifolds $$(f,G)$$ with $$f=x_1^n+cdots+x_N^n$$ and $$G=Sltimes G^d$$ , where $$Ssubseteq S_N$$ and $$G^d$$ is either the...     

Analytic types of plane curve singularities defined by weighted homogeneous polynomials

We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials , which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that either satisfies the same property as the above does or is homogeneous, then we prove easily that the weights of the above determine the topological type of and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in , which was already known. Also, as an application, it can be shown that for a given , where is a quasihomogeneous holomorphic function with an isolated singularity at the origin or with a positive integer , analytic types of isolated hypersurface singularities defined by are easily classified where is defined just as above.

     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homotheties and incidences

We consider problems involving rich homotheties in a set $S$ of $n$ points in the plane (that is, homotheties that map many points of $S$ to other points of $S$). By reducing these problems to incidence problems involving points and lines in ${\mathbb{R}}^3$, we are able to obtain refined and new bounds for the number of rich homotheties, and for the number of distinct equivalence classes, under homotheties, of $k$-element subsets of $S$, for any $k\ge 3$. We also discuss the extensions of these problems to three and higher dimensions.     

Askey-Wilson polynomials,kernel polynomials and association schemes

The eigenvalues of the classical association schemes are given by certain Askey-Wilson polynomials (or limiting cases). Kernel polynomials of the Askey-Wilson polynomials are also Askey-Wilson polynomials. We determine when both the Askey-Wilson polynomials and their kernels are the eigenvalues for known association schemes.