On the invariant theory of the Bézoutiant.
On the irreducibility of pure difference binomials.
On the Jacobian conjecture and the configurations of roots.
On the Jacobian criterion of formal smoothness.
We give a short proof of the Jacobian criterion of formal smoothness using the Lichtenbaum-Schlessinger cotangent complex.
On the Jacobian ideal of the binary discriminant.
Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...
On the Jung method in positive characteristic
Let be a germ of normal surface with local ring covering a germ of regular surface with local ring of characteristic . Given an extension of valuation rings birationally dominating , we study the existence of a new such pair of local rings birationally dominating , such that is regular and has only toric singularities. This is achieved when is defectless or when is equal to
On the Length of Faithful Modules over Artinian Local Rings.
On the length of the absolute Samuel stratum
On the lengths of bad sequences of monomial ideals over polynomial rings
We give bad (with respect to the reverse inclusion ordering) sequences of monomial ideals in two variables with Ackermannian lengths and extend this to multiple recursive lengths for more variables.
On the local uniformization problem
In this paper we give a short introduction to the local uniformization problem. This follows a similar line as the one presented by the second author in his talk at ALANT 3. We also discuss our paper on the reduction of local uniformization to the rank one case. In that paper, we prove that in order to obtain local uniformization for valuations centered at objects of a subcategory of the category of noetherian integral domains, it is enough to prove it for rank one valuations centered at objects...
On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ²
A complete characterization of the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² is given. Moreover, a characterization of a component of a polynomial automorphism of ℂ² (in terms of the Łojasiewicz exponent at infinity) is given.
On the maximal spectrum of commutative semiprimitive rings
The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R).
On the minimal reduction and multiplicity of .
On the minimaxness and coatomicness of local cohomology modules
Let be a commutative Noetherian ring, an ideal of and an -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and -minimaxness of local cohomology modules. We show that if is a minimax -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if is a nonnegative integer such that is a minimax -module for all and for all , then the set is finite. Also, if is minimax for...
On the module of homomorphisms on finitely generated projective modules
On the Mulitplicity of Zeros of Polynomials over Arbitrary Finite Dimensional K-Algebras.
On the multiplicity of .
On the Noether exponent
We obtain, in a simple way, an estimate for the Noether exponent of an ideal I without embedded components (i.e. we estimate the smallest number μ such that ).
On the Nonflatness of Analytic Tensor Product.
On the nontrivial solvability of systems of homogeneous linear equations over in ZFC
Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals , an arbitrary nonempty system of homogeneous -linear equations is nontrivially solvable in provided that each of its subsystems of cardinality less than is nontrivially solvable in ?