Current and Recent Research Projects

Projective curves through finite sets and rank decompositions of symmetric tensors with applications to Waring decompositions of quintic forms in four variables

In the preprint "Projective curves through finite sets and rank decompositions of symmetric tensors", joint with C. Bocci, L. Chiantini, A. Mazzon, and C. Peterson, we study minimality and identifiability of Waring decompositions of quintic forms in four variables. The generic rank of such forms is 14 by landmark result of Alexander and Hirschowitz. Chiantini, Ottaviani, and Vannieuwenhoven ( On generic identifiability of symmetric tensors of subgeneric rank ) prove that the generic tensor of subgeneric rank is identifiable. However, it is non-trivial to identify the conditions a tensor of subgeneric rank must satisfy to be identifiable. Our setup is to consider a set of distinct points A in projective three-space. Corresponding to this set of points is the collection of quintic forms F which may be written as linear combinations of fifth powers of the linear forms dual to the points of A (we say that A decomposes F). We assume that A is in general position, by which we mean that every subset of A imposes the maximum possible number of independent conditions on forms of degree d, for every positive degree d. If A has at most 12 points, then we show that general position is enough to imply the minimality and identifiability of any quintic form which A decomposes. On the other hand we show that if A consists of 13 points and F is decomposed by A, then if A is in general position and further does not lie on an integral quintic curve of genus one, then F has Waring rank 13. Further, if A does not lie on an aCM integral sextic curve of genus three, then F is identifiable. The final section of the paper is devoted to three algorithms which can be used to exclude the possibility of A lying on such curves. We implemented symbolic and numerical versions of these algorithms which can be found in the links below, along with worked examples.

Symbolic algorithms (Macaulay2):

Maximal Kruskal Ranks (Algorithm 1) . This tutorial shows how to execute the symbolic algorithm to check for maximal Kruskal ranks, which is Algorithm 1 in section 6 of our paper. In particular, we check that the running example in section 6 of the paper has maximal Kruskal ranks.
Verifying a set of points does not lie on an elliptic quintic curve (Algorithm 2) . This tutorial shows how to execute the symbolic algorithm to check that a set of thirteen points does not lie on an integral quintic curve of genus one, which is Algorithm 2 in section 6 of our paper. In particular, we check that the running example in section 6 of the paper does not lie on an integral quintic curve of genus one.
Jacobian Algorithm (Algorithm 3) . This tutorial shows how to execute the Jacobian algorithm to verify that a set of thirteen points does not lie on an integral sextic curve of genus three, which is Algorithm 3 in section 6 of our paper. In particular, we check that the running example in section 6 of the paper does not lie on an integral sextic curve of genus three.

Numerical algorithms (Sage/Python):

Koszul multi-Rees algebras of L-Borel ideals

In the preprint Koszul multi-Rees algebras of L-Borel ideals , joint with Babak Jabbar Nezhad, we show that the multi-Rees algebra of a collection of principal L-Borel ideals satisfying a certain incidence condition is Koszul, Cohen-Macaulay, and normal. We do this by exhibiting a Grobner basis of quadrics with squarefree lead terms for the defining equations of the multi-Rees algebra with respect to Lexicographic order. As a corollary we obtain that the multi-Rees algebra of principal Borel ideals (and its multi-fiber ring) is Koszul, which has been expected for some time. Below is some Macaulay2 code which accompanies the paper. The file BorelSort.m2 contains an implementation of the Borel Sort algorithm, which is Algorithm 1 in our preprint. The file MultiReesCode.m2 contains code to verify several examples in the paper. To execute these in Macaulay2, first download both scripts and then modify the beginning of the MultiReesCode.m2 script to reflect the path in your local directory where BorelSort.m2 is contained. Then copy and paste the commands from MultiReesCode.m2 into your Macaulay2 session.

BorelSort.m2
MultiReesCode.m2

Lower bounds for splines on vertex stars and tetrahedral complexes

In the preprint A lower bound for splines on tetrahedral vertex stars , joint with Nelly Villamizar, we give a lower bound for the dimension of homogeneous splines on vertex stars. In a follow-up preprint, A lower bound for the dimension of tetrahedral splines in large degree (also joint with Nelly Villamizar), we give a lower bound on the dimension of tetrahedral splines in large degree.

The file BoundFunctions.m2 defines the functions to produce bounds for closed vertex stars, open vertex stars, and tetrahedral complexes.
The file VertexStars.m2 uses the functions defined in BoundFunctions.m2 to create the data which can be found in Tables 1 and 2 of the preprint 'A lower bound for splines on tetrahedral vertex stars.' The file TrivariateSplines.m2 uses the functions defined in BoundFunctions.m2 to create the data for the examples in 'A lower bound for the dimension of tetrahedral splines in large degree.'

BoundFunctions.m2
VertexStars.m2
TrivariateSplines.m2

The apolar algebra of a product of linear forms

In the preprint The apolar algebra of a product of linear forms , joint with Zach Flores and Chris Peterson, we give some results concerning the structure of the apolar algebra of a multi-arrangement (that is, the apolar algebra of a homogeneous polynomial which factors completely as a product of linear forms which are not necessarily distinct). Section 5 of this paper, and Theorem 5.1 in particular, reports on the results of a computation we carried out using the computer algebra system Bertini. The files below contain the code which we used to first generate the equations yielding the polynomials in Theorem 5.1 and then verify their properties which we give in Tables 1 and 2 of this paper.

The file SixLines.m2 below contains code which produces the catalecticant rank equations which we then ask Bertini to solve.
The file input_cubic_annihilator is a file which Bertini can run to produce the forms listed in Theorem 5.1 (after some not insignificant post-processing).
The file SixLinesRank.m2 contains code to verify that the cubics listed in Table 1 do indeed annihilate the forms listed in Theorem 5.1. Furthermore this file verifies that the ideal generated by forms of degree at most three in the apolar ideals of each of the forms in Theorem 5.1 are indeed saturated and, for all but one of the forms, reduced.
The file SageApolar.txt contains code (to be executed in Sage) to produce the cubics which annihilate the two forms in Theorem 5.1 which are not defined over the rationals.
Finally, the file VerifyWaringRank.txt (also to be executed in Sage) verifies that the Waring decompositions claimed in Table 2 are indeed correct.

SixLines.m2
input_cubic_annihilator
SixLinesRank.m2
SageApolar.txt
VerifyWaringRank.txt

Homological characterizations for freeness of multi-arrangements

In the paper A homological characterization for freeness of multi-arrangements , the methods used in the project "free multiplicities on braid arrangements" below are generalized to arbitrary arrangements. We construct a cochain complex, which we'll call the derivation complex, whose first cohomology is the module D(A,m) of multi-derivations on the arrangement A and whose higher cohomologies vanish if and only if (A,m) is free. In the linked file DerivationComplex.m2, this cochain complex is constructed using the command derivationComplex. The derivation complex is the third (co)chain complex in a short exact sequence with two other cochain complexes. The middle cochain complex, which we'll call the formality complex, is constructed by the command formalityComplex in the linked file DerivationComplex.m2. It has cohomologies which vanish if and only if A is k-formal in the sense of Brandt and Terao. The formality complex surjects naturally onto the derivation complex; the kernel of this map of cochain complexes is another cochain complex, which is constructed with the command kernelDerivationComplex in the linked file DerivationComplex.m2.

To use the functions in the file DerivationComplex.m2, copy and paste the text from the file and save it under the name DerivationComplex.m2. Then load the file in a Macaulay2 session. The file DerivationComplexExamples.m2 contains examples of how to use the code in DerivationComplex.m2. It is written to be executed line by line in Macaulay2. The file DerivationComplexVerify.m2 has the code to verify examples in the paper "A homological characterization for freeness of multi-arrangements."

DerivationComplex.m2
DerivationComplexExamples.m2
DerivationComplexVerify.m2

Free Multiplicities on Braid and Graphic Arrangements

This project is centered on using techniques from spline theory to study freeness of multi-arrangements, particularly braid arrangements and their sub-arrangements (known as graphic arrangements). It has long been known that the module of multi-derivations on an arrangement and the module of splines on a polytopal complex have a number of similarities. Moreover, multi-derivations on the braid arrangement co-incide with the module of splines on the so-called 'Alfeld split.' Schenck recently exploited this relationship to prove dimension formulas for the space of splines on the Alfeld split, which were conjectured by Sorokina and Foucart.
In the paper Generalized splines and graphic arrangements , we show that multi-derivations on sub-arrangements of the braid arrangement (alsoknown as graphic arrangements) also co-incide with certain spline modules. The Billera-Schenck-Stillman chain complex for splines then yields new homological obstructions for freeness of graphic multi-arrangements. In Free and non-free multiplicities on the A_3 braid arrangement (joint with Francisco, Mermin, and Schweig) we work out precisely what these obstructions are for the A_3 braid arrangement. Building on previous work of Abe, Nuida, Numata, Wakefield, and Terao, we complete the classification of free multiplicities on the A_3 braid arrangement.
In a third paper, Inequalities for free multi-braid arrangements , we partially extend the A_3 classification to higher braid arrangements. On a large cone containing the constant multiplicities, we show that the only free multiplicities are those identified by Abe, Nuida, and Numata. A shortened version of this paper has appeared in the Proceedings of the Japan Academy, Series A; since details of some tedious computations for Proposition 5.5 were ommitted in this submission, the reader may find more details for that particular computation here: Supplemental computations for sigma cycles, mountains, and hills .

Semialgebraic Splines

Most work on splines is restricted to subdivisions where the edges are lines. However, some work has been done by RenHong Wang and Peter Stiller in the context of more general partitions, where lines are replaced by irreducible algebraic curves. Recent work of Davydov, Kostin, and Saeed in the context of isogeometric analysis suggests that such partitions may be of increasing use in the finite element method.
In the paper Semialgebraic splines , joint with Frank Sottile and Lanyin Sun, the dimension of the space of splines of degree at most d on a subdivision (in the plane) consisting of irreducible curves meeting at the origin is computed. Frank Sottile has put together a nice array of pictures and code explaining our results, which may be found on his website, here .