arXiv subjects
Publications and source records attributed to Szymon Brzostowski.
The paper titled "Cremona problem in dimension 2" by W. Bartenwerfer presented a flawed attempt at proving the Jacobian Conjecture. Our aim is to provide a thorough analysis of the author's approach, highlighting the errors that were made in the process.
We prove the constancy of the Łojasiewicz exponent in non-degenerate $μ$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.
We prove that if two plane curve singularities are equisingular, then they are topologically equivalent. The method we will use is P.~Fortuny~Ayuso's who proved this result for irreducible plane curve singularities.
According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $μ(f)$ is equal to the Newton number $ν(Γ_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $Γ_+ (f)$. We give a simple condition characterising, in terms of $Γ_+ (f)$ and $Γ_+ (g)$, the equality $ν(Γ_{+}(f)) = ν(Γ_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $Γ_+ (f) \subset Γ_+ (g)$. This is a complete solution to an Arnold's problem (1982-16) in this case.
Let $k$ be an algebraically closed field of any characteristic. We apply the Hamburger-Noether process of successive quadratic transformations to show the equivalence of two definitions of the Łojasiewicz exponent $\mathfrak{L}(\mathfrak{a})$ of an ideal $\mathfrak{a}\subset k[[x,y]]$.
Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities in the case deformations are non-degenerate.
Let $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local Łojasiewicz exponent $\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a quasihomogeneous isolated singularity $f$, we generalize a formula for $\mathcal{L}_{0}(f)$ of Krasiński, Oleksik and Płoski ([KOP09]) from $3$ to $n$ dimensions. This was previously announced in [TYZ10], but as a matter of fact it has not been proved correctly there, as noticed by the AMS reviewer T. Krasiński. As a consequence of our result, we get that the Łojasiewicz exponent is invariant in topologically trivial families of singularities coming from a quasihomogeneous germ. This is an affirmative partial answer to Teissier's conjecture. References [KOP09] Tadeusz Krasiński, Grzegorz Oleksik and Arkadiusz Płoski. The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity. Proc. Amer. Math. Soc., 137(10):3387-3397, 2009. [TYZ10] Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo. Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity. Proc. Amer. Math. Soc., 138(11):3975-3984, 2010.
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)).$ We prove that for the singularities in the $X_9$ singularity class their jumps are equal to 2.