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Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (C n , 0) → (C , 0) . We introduce a new derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let I = (f , ∂ f ∂ x 1 , … , ∂ f ∂ x n ) and Δ (I) : = { g ∣ g , ∂ g ∂ x 1 , … , ∂ g ∂ x n ∈ I } , then A Δ (V) : = O n / Δ (I) and L Δ (V) : = Der (A Δ (V) , A Δ (V)) . Their dimensions as a complex vector space are denoted as β (V) and δ (V) , respectively. δ (V) is a new invariant of singularities. In this paper we study the new local algebra A Δ (V) and the derivation Lie algebra L Δ (V) , and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimation conjectures for β (V) and δ (V) when (V, 0) are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities. Lastly, we provide an application of β (V) and δ (V) to distinguishing contact classes of singularities. [ABSTRACT FROM AUTHOR]

Following Shokurov's idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new invariant to isolated complete intersection singularities. We use this new invariant to obtain two characterization theorems for contact simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps. [ABSTRACT FROM AUTHOR]

For i : Z ↪ X a closed immersion of smooth varieties, we study how the V-filtration along Z and the Hodge filtration on a mixed Hodge module M on X interact with each other. We also give a formula for the functors i ∗ , i ! in terms of this V-filtration. As applications, we obtain results on the Hodge filtration of monodromic mixed Hodge modules and we give a Hodge theoretic proof of Skoda's theorem on multiplier ideals. Finally, we use the results to study the Fourier–Laplace transform of a monodromic mixed Hodge module. [ABSTRACT FROM AUTHOR]

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in C n which is Lipschitz regular at infinity must be an affine linear subspace of C n . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here. [ABSTRACT FROM AUTHOR]

We study the motivic and ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair. [ABSTRACT FROM AUTHOR]

In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula. [ABSTRACT FROM AUTHOR]

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores. [ABSTRACT FROM AUTHOR]

We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of the Kronecker solver and a new improved algorithm for fast multivariate modular composition. [ABSTRACT FROM AUTHOR]

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions. [ABSTRACT FROM AUTHOR]

For an analytic set V in K n , K = R or C , which contains the origin 0 ∈ K n , the geometric tangent cone of V at 0 is the set of vectors in K n which are the limits of secant lines passing through the origin and non-zero sequences in V that converge to the origin; the algebraic tangent cone of V at 0 is the algebraic set defined by the ideal generated by the initial forms of all analytic functions f whose germs at 0 are in the ring of germs of analytic functions in K n about 0 which vanish on the germ of V at 0. In this paper, we give some characterization for the geometric tangent cone and compare these two tangent cones of V at 0. In particular, we characterize the geometric tangent cone of V at 0 via the so-called Ł ojasiewicz exponent of an analytic map germ along a line and compute this number in some special cases. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R / I (t) = dim R - pd I (t) - 1 , where I (t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R / I (t) is non-increasing if I is a squarefree monomial ideal. We show that depth R / I (t) is almost non-increasing in the sense that depth R / I (s) ≥ depth R / I (t) for all s ≥ 1 and t ∈ E (s) , where E (s) = ⋃ i ≥ 1 { t ∈ N | i (s - 1) + 1 ≤ t ≤ i s } (which contains all integers t ≥ (s - 1) 2 + 1 ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R / I (s) < depth R / I (t) for t ∉ E (s) , which gives a negative answer to the above question. Another open question asks whether the function depth R / I (t) is always constant for t ≫ 0 . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I (t) is integrally closed for t ≫ 0 (e.g. if I is a squarefree monomial ideal), then depth R / I (t) is constant for t ≫ 0 with lim t → ∞ depth R / I (t) = dim R - dim ⊕ t ≥ 0 I (t) / m I (t). Our last result (which is the main contribution of this paper) shows that for any positive numerical function ϕ (t) which is periodic for t ≫ 0 , there exist a polynomial ring R and a homogeneous ideal I such that depth R / I (t) = ϕ (t) for all t ≥ 1 . As a consequence, for any non-negative numerical function ψ (t) which is periodic for t ≫ 0 , there is a homogeneous ideal I and a number c such that pd I (t) = ψ (t) + c for all t ≥ 1 . [ABSTRACT FROM AUTHOR]

The Milnor number of an isolated hypersurface singularity, defined as the codimension μ(f) of the ideal generated by the partial derivatives of a power series f that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation f representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series f in two indeterminates over the complex numbers, it was shown by Milnor that μ(f) always coincides with the conductor c(f) of the semigroup of values S(f) of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup S(f) is tame, that is, the characteristic of the field does not divide any of its minimal generators. [ABSTRACT FROM AUTHOR]

In this note, we show that the free generators of the Mishchenko-Fomenko subalgebra of a complex reductive Lie algebra, constructed by the argument shift method at a regular element, form a regular sequence. This result was proven by Serge Ovsienko in the type A at a regular and semisimple element. Our approach is very different, and is strongly based on geometric properties of the nilpotent bicone. [ABSTRACT FROM AUTHOR]

Given an analytic function germ f: ( X, 0) → C on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f (0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras. [ABSTRACT FROM AUTHOR]

We develop a theory of 'ad hoc' Chern characters for twisted matrix factorizations associated to a scheme X, a line bundle $$\mathcal {L}$$ , and a regular global section $$W \in \Gamma (X, \mathcal {L})$$ . As an application, we establish the vanishing, in certain cases, of $$h_c^R(M,N)$$ , the higher Herbrand difference, and, $$\eta _c^R(M,N)$$ , the higher codimensional analogue of Hochster's theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated R-modules. Specifically, we prove such vanishing if $$R = Q/(f_1, \dots , f_c)$$ has only isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0, the $$f_i$$ 's form a regular sequence, and $$c \ge 2$$ . Such vanishing was previously established in the general characteristic, but graded, setting in Moore et al. (Math Z 273(3-4):907-920, 2013). [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over $$\mathbb {Q}_p$$ , and let $$\Gamma $$ be a compact open subgroup of $$G(\mathbb {Q}_p)$$ . We relate the asymptotic representation theory of $$\Gamma $$ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both: For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. [ABSTRACT FROM AUTHOR]

Based on Tiankai Liu's PhD thesis [], we give a complete classification of local topological types of singularities with two Newton pairs on rational unicuspidal complex projective plane curves. [ABSTRACT FROM AUTHOR]

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 regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu's result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets. [ABSTRACT FROM AUTHOR]

It is well known that the main approaches of the analytical solving of the elasticity mixed plane problems for a semi-strip are based on the different representations of the equilibrium equations' solutions: the representations through the harmonic and by harmonic functions, through the stress function, Fadle-Papkovich functions and so on. The main shortcoming of these approaches is connected with the fact that to obtain the expression for the real mechanical characteristics, one should execute additional operations, not always simple ones. The approach that is proposed in this paper allows the direct solution of the equilibrium equations. With the help of the matrix integral transformation method applied directly to the equilibrium equations, the initial boundary problem is reduced to a vector boundary problem in the transformation's domain. The use of matrix differential calculations and Green's matrix function leads to the exact vector solution of the problem. Green's matrix function is constructed in the form of a bilinear representation which simplifies the calculations. The method is demonstrated by the solving of the thermoelastic problem for the semi-strip. The zones and conditions of the strain stress occurrence on the semi-strip's lateral sides, important to engineering applications, are investigated. [ABSTRACT FROM AUTHOR]

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes. [ABSTRACT FROM AUTHOR]

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230-256, ) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129-199, ). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ . [ABSTRACT FROM AUTHOR]

Let $$X$$ be a smooth variety over an algebraically closed field of characteristic $$p > 0, Z$$ a smooth divisor, and $$j: U=X {\setminus } Z \rightarrow X$$ the natural inclusion. We introduce in an axiomatic way the notion of a $$V$$ -filtration on unit $$F$$ -crystals and prove such axioms determine a unique filtration. It is shown that if $$\mathcal M $$ is a tame unit $$F$$ -crystal on $$U$$ , then such a $$V$$ -filtration along $$Z$$ exists on $$j_*\mathcal M $$ . The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton-Kisin Riemann-Hilbert correspondence. A few applications to $$\mathbb A ^1$$ and gluing are then discussed. [ABSTRACT FROM AUTHOR]

We describe a natural decomposition of a normal complex surface singularity ( X, 0) into its 'thick' and 'thin' parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of ( X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants. [ABSTRACT FROM AUTHOR]

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants c > 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals. [ABSTRACT FROM AUTHOR]

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and François Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let $$k$$ be an arbitrary field of characteristic zero, and let $$X$$ be a smooth quasi-projective $$k$$ -scheme. Precisely, we show that, in the Grothendieck group of constructible étale motives, the image of the nearby motive associated with a flat morphism of $$k$$ -schemes $$f:X\rightarrow \mathbb A ^1_k$$ , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with $$f$$ . In particular, it provides a realization of the motivic Milnor fiber of $$f$$ in the “non-virtual” motivic world. [ABSTRACT FROM AUTHOR]

We prove that for N≥4, all smooth hypersurfaces of degree N in ℙ are birationally superrigid. First discovered in the case N=4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals. [ABSTRACT FROM AUTHOR]

Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let $R^{1/p^{e}}$ be the ring of p-th roots of elements of R for e∈ℕ, and let a denote the maximal rank of a free R-module appearing in a direct sum decomposition of $R^{1/p^{e}}$. We show the existence of the limit $s(R) := \lim_{e \to\infty} \frac{a_{e}}{p^{ed}}$, called the F-signature of R. This invariant-which can be extended to all local rings in prime characteristic-was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391-404, ) and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of arbitrary finite quotient singularities. [ABSTRACT FROM AUTHOR]

For a real square-free multivariate polynomial F, we treat the general problem of finding real solutions of the equation F=0, provided that the real solution set { F=0} is compact. We allow that the equation F=0 may have singular real solutions. We are going to decide whether this equation has a non-singular real solution and, if this is the case, we exhibit one for each generically smooth connected component of { F=0}. We design a family of elimination algorithms of intrinsic complexity which solves this problem. In the worst case, the complexity of our algorithms does not exceed the already known extrinsic complexity bound of ( nd) for the elimination problem under consideration, where n is the number of indeterminates of F and d its (positive) degree. In the case that the real variety defined by F is smooth, there already exist algorithms of intrinsic complexity that solve our problem. However, these algorithms cannot be used in case when F=0 admits F-singular real solutions. An elimination algorithm of intrinsic complexity presupposes that the polynomial F is encoded by an essentially division-free arithmetic circuit of size L (i.e., F can be evaluated by means of L additions, subtractions and multiplications, using scalars from a previously fixed real ground field, say ℚ) and that there is given an invariant δ( F) which (roughly speaking) depends only on the geometry of the complex hypersurface defined by F. The complexity of the algorithm (measured in terms of the number of arithmetic operations in ℚ) is then linear in L and polynomial in n, d and δ( F). In order to find such a geometric invariant δ( F), we consider suitable incidence varieties which in fact are algebraic families of dual polar varieties of the complex hypersurface defined by F. The generic dual polar varieties of these incidence varieties are called bipolar varieties of the equation F=0. The maximal degree of these bipolar varieties then becomes the essential ingredient of our invariant δ( F). [ABSTRACT FROM AUTHOR]

A complete classification of the individual types of singular points is given for irreducible real sextic curves. This classification is derived by using the computer algebra system Maple. There are 191 types of singular points for real irreducible sextic curves. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. A significant portion of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple. [ABSTRACT FROM AUTHOR]