31 research outputs found
Strong Algebras and Radical Sylvester-Gallai Configurations
In this paper, we prove the following non-linear generalization of the
classical Sylvester-Gallai theorem. Let be an algebraically closed
field of characteristic , and be a set of irreducible homogeneous polynomials of
degree at most such that is not a scalar multiple of for . Suppose that for any two distinct , there is such that . We prove that such radical SG
configurations must be low dimensional. More precisely, we show that there
exists a function , independent of
and , such that any such configuration must
satisfy
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
Evidence, Proofs, and Derivations
The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas
Idéaux de preuve : explication et pureté
Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof
Approximate algebraic structure
We discuss a selection of recent developments in arithmetic combinatorics
having to do with ``approximate algebraic structure'' together with some of
their applications.Comment: 25 pages. Submitted to Proceedings of the ICM 2014. This version may
be longer than the published one, as my submission was 4 pages too long with
the official style fil
Lifting matroid divisors on tropical curves
Tropical geometry gives a bound on the ranks of divisors on curves in terms
of the combinatorics of the dual graph of a degeneration. We show that for a
family of examples, curves realizing this bound might only exist over certain
characteristics or over certain fields of definition. Our examples also apply
to the theory of metrized complexes and weighted graphs. These examples arise
by relating the lifting problem to matroid realizability. We also give a proof
of Mn\"ev universality with explicit bounds on the size of the matroid, which
may be of independent interest.Comment: 27 pages, 7 figures, final submitted version: several proofs
clarified and various minor change