arXiv subjects
Publications and source records attributed to Adam Różycki.
It is well known that in the Noetherian local ring with infinite residue field the reduction of $\mm$-primary ideal may be given in the form of a sufficiently general linear combination of its generators. In the paper we give a condition for the existence of such reduction in terms of the sum of degrees of the ideal fiber cone prime divisors in the case of any Noetherian local ring.
The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: \emph{Let X be a nonsingular closed subvariety of $\mathbb{P}^n_k$, where $k$ is an algebraically closed field. Then there exists a hyperplane $H\subset \mathbb{P}^n_k$ not containing $X$ and such that the scheme $H\cap X$ is regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system $|H|$ considered as a projective space. } We will show that one can effectively indicate a finite family of hyperplanes $H$ such that at least one of them satisfies the assertion of the Bertini theorem. As an application of the method used in the proof we will give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.
We give an algorithm to compute the Lê numbers of (the germ of) a Newton non-degenerate complex analytic function $f\colon(\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ in terms of certain invariants attached to the Newton diagram of the function $f+z_1^{α_1}+\cdots +z_d^{α_d}$, where $d$ is the dimension of the critical locus of $f$ and $α_1,\ldots, α_d$ are sufficiently large integers. This is a version for non-isolated singularities of a famous theorem of A. G. Kouchnirenko. As a corollary, we obtain that Newton non-degenerate functions with the same Newton diagram have the same Lê numbers.