Search

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order., Comment: 19 pages

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,...,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety., Comment: 24 pages

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements

For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces of $X$ are embedded into their spectra as topological spaces., 20 pages

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties., Accepted to Transactions of the London Mathematical Society

We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar., Comment: 28 pages. Final Final version

We introduce a "workable" notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic B\'ezout Inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators. We compute probabilistically the degree of an equidimensional ideal.

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$., Comment: Final version, to appear in Journal of Algebra; minor changes, unabbreviated title, updated references

We prove that under suitable graded and local hypothesis, a formally unramified algebra over a field must be reduced. We detail examples, including one due to Gabber, to show that it is not possible to generalize these results further., Comment: 8 pages; comments are welcome

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”.

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show the above is true when replacing irreducible components by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras., 26 pages. Updated references, typos, metadata. Actually final version

In this article, we establish a motivic analog of an enumeration result of James-Thomas on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank $d$ over a smooth affine $k$-algebra $A$ of dimension $d$, recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank $d-1$ if the base field $k$ is algebraically closed. We also explore the cancellation properties of symplectic vector bundles., Comment: New version, 37 pages. Added a cancellation result for symplectic vector bundles and exposition improved. Comments are welcome!

The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let $C_1$ and $C_2$ be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of $C_1$ and the dual $C_2^\perp$. Moreover, we give an explicit description of a generator matrix of $C_2^\perp$ in terms of that of $C_1$ and coefficients of indicator functions. For Reed--Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed--Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code., The previous version has a typo on the statement of Theorem 5.4. The published version can be found here: https://rdcu.be/cjon5

Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity $3$. We classify graphs whose parity binomial edge ideals have pure resolution., 10 pages, Suggestions and comments are welcome. arXiv admin note: text overlap with arXiv:1911.10388

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed., Comment: To appear in Trans. Am. Math. Soc

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which the curve defines an extremal class in the effective cone. We give a complete classification of these blowups into Mori Dream Spaces and non-Mori Dream Spaces. Our approach greatly simplifies previous proofs, avoiding positive characteristic methods and higher cohomology., 21 pages, 3 figures

In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally $F$-split projective variety., Comment: Fixed errors in the proof of Lemma 3.7 and in Example 4.11 of previous version

It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements., Comment: To appear in Discrete & Computational Geometry. arXiv admin note: substantial text overlap with arXiv:1707.08449, arXiv:1703.09402

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations., Comment: 32 pages. To appear in Foundations of Computational Mathematics

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer [6], and by Wall and Mattei [11, 13]., The first and third authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2015-69135-P, Generalitat de Catalunya 2017SGR-932 projects, and they are with the Barcelona Graduate School of Mathematics (BGSMath), through the project MDM-2014-0445. The second and fourth authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2016-76868-C2-1-P

In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects. In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is approximable whenever $X$ is a quasicompact separated scheme. As corollaries of this (seemingly technical) statement one can prove striking improvements on old theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much smaller) categories $D^{\text{perf}}(X)$ and $D^b_{\text{coh}}(X)$., Comment: Final pre-publication version

Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers., Comment: v2: added Subsections 6.1 (some remarks about compatibility with char. 0) 6.5 (other characterizations of Bernstein-Sato roots) and 6.6 (examples). Relaxed assumptions on R. 33 pages

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2., 26 pages; the Macaulay2 package "ApolarLowRank.m2" is available in the ancillary files

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method., 16 pages

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the bipartite complement of $G$ is a tree of diameter at most $3$. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions., 20 pages, comments welcome

In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires some delicate studies., Comment: To appear in J. Algebra

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give a new proof that irreducible arrangements with a generic hyperplane are totally non-free. In another application, we show that an irreducible multiarrangement of rank $3$ with at least two locally heavy hyperplanes is not free., Comment: 16 pages, 1 figure. arXiv admin note: text overlap with arXiv:1603.05803

Given a finite order ideal $\mathcal O$ in the polynomial ring $K[x_1,\dots, x_n]$ over a field $K$, let $\partial \mathcal O$ be the border of $\mathcal O$ and $\mathcal P_{\mathcal O}$ the Pommaret basis of the ideal generated by the terms outside $\mathcal O$. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $\partial\mathcal O$-marked sets (resp. bases) and $\mathcal P_{\mathcal O}$-marked sets (resp. bases). We prove that a $\partial\mathcal O$-marked set $B$ is a marked basis if and only if the $\mathcal P_{\mathcal O}$-marked set $P$ contained in $B$ is a marked basis and generates the same ideal as $B$. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $\partial\mathcal O$-marked bases and $\mathcal P_{\mathcal O}$-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gr\"obner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in smaller affine spaces. Furthermore, we observe that Pommaret marked schemes give an open covering of punctual Hilbert schemes. Several examples are given along all the paper., Comment: 17 pages; presentation improved, some references added

Let $$(R,{\mathfrak {m}},\mathbb {k})$$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $$\mathbb {k}$$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $$\Omega _R$$ is a torsion-free R module. We give new cases of this conjecture by extending works of Guttes (Arch Math 54:499–510, 1990) and Cortinas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of $${\text {Hom}}_R({\mathfrak {m}},{\mathfrak {m}})$$ and constructing enough torsion in $$\Omega _S$$ , enabling us to pull back a nontrivial torsion to $$\Omega _R$$ .

We give examples of surfaces which are Ulrich-wild, i.e. that support families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$., Comment: 15 pages. Several typos have been fixed. The proof of Theorem 1.2 has been simplified. The final version will appear in Proceedings of the A.M.S

We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus., Comment: Final version, to appear in Inventiones Math. The statements concerning positive characteristic have been removed and will be published separately. This submission supersedes arXiv:1910.06200 and arXiv:1907.07553

In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements., To appear in the Journal of Algebra

Hartshorne conjectured that a smooth, codimension c subvariety of n-dimensional projective space must be a complete intersection, whenever c is less than n/3. We prove this in the special case when n is much larger than the degree of the subvariety. Similar results were known in characteristic zero due to Hartshorne, Barth-Van de Ven, and others. Our proof is field independent and employs quite different methods from those previous results, as we connect Hartshorne's Conjecture with the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture., Comment: 5 pages

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular library realclassify.lib., 31 pages, 5 figures, 1 table, improvements in the algorithms

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $. In this paper we exhibit an algorithm to compute generators for $\left[ A_{1},\cdots ,A_{M};C^{m}\right] $., 47 pages, see also the related article "Solutions to a System of Equations for $C^m$ Functions"

What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws., Comment: 12 pages

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of J{\o}rgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology., Comment: 41 pages, final version, to appear in Transactions of the AMS

For a planar simplicial complex Delta contained in R^2, Schumaker proved that a lower bound on the dimension of the space C^r_k(Delta) of planar splines of smoothness r and polynomial degree at most k on Delta is given by a polynomial P_Delta(r,k), and Alfeld-Schumaker showed this polynomial gives the correct dimension when k >= 4r+1. Examples due to Morgan-Scott, Tohaneanu, and Yuan show that the equality dim C^r_k(Delta) = P_Delta(r,k) can fail when k = 2r or 2r+1. We prove that the equality dim C^r_k(Delta)= P_Delta(r,k) cannot hold in general for k, v1: 4 pages, 1 figure. v2: minor stylistic changes. v3: added additional algebraic background

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0., Agencia Estatal de Investigación, Ministerio de Economía y Competitividad, Austrian Science Fund

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular we present examples of polynomials $g$ and $f$ such that there is no differential operator raising $g/f$ to its $p$th power., 14 pages, comments are welcome

Let $k$ be a field of positive characteristic. Building on the work of the second named author, we define a new class of $k$-algebras, called diagonally $F$-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth $k$-algebras. We also show that this class contains certain singular algebras, such as the affine cone over $\mathbb{P}^r_{k} \times \mathbb{P}^s_{k}$, when $k$ is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over $\mathbb{P}^r_{\mathbb{C}} \times \mathbb{P}^s_{\mathbb{C}}$ and more generally for complex varieties of diagonal $F$-regular type., Comment: 22 pages. Comments welcome

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to those of Demailly-Ein-Lazarsfeld and Hara-Yoshida. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras. We also show that our subadditivity containment is sharper than ones shown previously by Takagi and Eisenstein. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas of Takagi and Eisenstein to show that the adjoint ideal $\mathscr{J}_X(A, Z) $ reduces mod $p$ to Takagi's adjoint test ideal, even when the ambient space is singular, provided that $A$ is regular at the generic point of $X$. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting., Comment: Many changes big and small. Most notably, the proofs of theorems 3.12 and 5.15 were corrected in response to referee feedback. I also cleaned up the notation in the proof of 5.15

This note describes a package for computing seminormalization of rings within Macaulay2., Comment: 6 pages, comments welcome, the latest version of the package is available in https://github.com/kschwede/M2/blob/master/M2/Macaulay2/packages/Seminormalization.m2

We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 \times 3$ permanent is at least $14$., 17 pages. v2: clarifications and additional references. To appear in Linear Algebra Appl

We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes., Comment: 19 pages, 1 figure, comments welcome

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances., Comment: 26 pages, added a discussion on the monic rank for reducible cones

We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective families over the integers., 20 pages. Minor revisions to exposition