Related papers: On Zariski's multiplicity problem at infinity

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz…

Let $K$ be a number field and $f_1,\ldots,f_s\in K[x_1,\ldots,x_n]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontrivial zero in $K^n$. Apart from some small degrees, the…

In this paper, we identify two fractals if and only if they are biLipschitz equivalent. Fix ratio $r,$ for dust-like graph-directed sets with ratio $r$ and integer characteristics, we show that they are rigid in the sense that they are…

The graded cellularity of Libedinsky Double Leaves, which form a basis for the endomorphism ring of the Bott_Samelson_Soergel bimodules, allows us to view the Kazhdan_Lusztig polynomials as graded decomposition numbers. Using this point of…

Let $\Lambda$ be an artin algebra. We obtain that $\Lambda$ is syzygy-finite when the radical layer length of $\Lambda$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the…

In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2-dimensional SFTs (i.e. tilings). To be more precise, we prove that given any \pizu subset $P$ of $\{0,1\}^\NN$ there is a SFT $X$…

Let $k$ be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, $k$ could be an imaginary quadratic number field. Brauer proved that if $f_1, \ldots, f_r$ are…

The following ``Key Lemma'' plays an important role in Parusinski's work on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials…

We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral…

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional…

In this paper we show that there is an infinite number of finite groups with two relative subgroup commutativity degrees. Also, we indicate a sufficient condition such that a finite group has at least three relative subgroup commutativity…

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings, with perfect residue fields. We give a…

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric…

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some…

Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$})…

We study two modifications of the Post Correspondence Problem (PCP), namely 1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and 2) the conjugate version, where…

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$…

For a finite-dimensional algebra {\Lambda}, we establish an explicit bijection between widely generated torsion(-free) classes and semibricks in mod {\Lambda}. Using the kappa order on the lattice of torsion classes with canonical join…

We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $\Theta$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the…

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We…