Strong Algebras and Radical Sylvester-Gallai Configurations

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}=\{F_1,\cdots,F_m\} \subset \mathbb{K}[x_1,\cdots,x_N]$ be a set of irreducible homogeneous polynomials of degree at most $d$ such that $F_i$ is not a scalar multiple of $F_j$ for $i\neq j$. Suppose that for any two distinct $F_i,F_j\in \mathcal{F}$, there is $k\neq i,j$ such that $F_k\in \mathrm{rad}(F_i,F_j)$. We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function $\lambda : \mathbb{N} \to \mathbb{N}$, independent of $\mathbb{K},N$ and $m$, such that any such configuration $\mathcal{F}$ must satisfy $$\dim (\mathrm{span}_{\mathbb{K}}{\mathcal{F}}) \leq \lambda(d).$$ Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,OS22]. Our result takes us one step closer towards the first deterministic polynomial time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4 circuits of bounded top and bottom fanins. Our result, when combined with the Stillman uniformity type results of [AH20a,DLL19,ESS21], yields uniform bounds for several algebraic invariants such as projective dimension, Betti numbers and Castelnuovo-Mumford regularity of ideals generated by radical SG configurations.Comment: 62 pages. Comments are welcome

Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. - Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique. - Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=. We show that in the special case when I=, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space. - Given f(X)in F[X] by an arithmetic circuit and I=, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=

Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture