Superpolynomial lower bounds against low-depth algebraic circuits

N Limaye, S Srinivasan, S Tavenas - Communications of the ACM, 2024 - dl.acm.org
An Algebraic Circuit for a multivariate polynomial P is a computational model for constructing
the polynomial P using only additions and multiplications. It is a syntactic model of …

Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits

P Dutta, P Dwivedi, N Saxena - arXiv preprint arXiv:2304.11325, 2023 - arxiv.org
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-
$4 $ reduction result by Agrawal and Vinay (FOCS 2008) has made PIT for depth-$4 …

Hitting sets and reconstruction for dense orbits in vp_ {e} and ΣΠΣ circuits

D Medini, A Shpilka - 36th Computational Complexity …, 2021 - drops.dagstuhl.de
In this paper we study polynomials in VP_ {e}(polynomial-sized formulas) and in ΣΠΣ
(polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL^{aff} …

Hitting sets for orbits of circuit classes and polynomial families

C Saha, B Thankey - ACM Transactions on Computation Theory, 2021 - dl.acm.org
The orbit of an n-variate polynomial f (x) over a field is the set. The orbit of a polynomial f is a
geometrically interesting subset of the set of affine projections of f. Affine projections of …

Polynomial time deterministic identity testing algorithm for Σ[3]ΠΣΠ[2] circuits via Edelstein–Kelly type theorem for quadratic polynomials

S Peleg, A Shpilka - Proceedings of the 53rd Annual ACM SIGACT …, 2021 - dl.acm.org
In this work we resolve conjectures of Beecken, Mitmann and Saxena [BMS13] and Gupta
[Gupta14], by proving an analog of a theorem of Edelstein and Kelly for quadratic …

A generalized Sylvester–Gallai-type theorem for quadratic polynomials

S Peleg, A Shpilka - Forum of Mathematics, Sigma, 2022 - cambridge.org
In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials
that takes us one step closer to obtaining a deterministic polynomial time algorithm for …

Sylvester-Gallai type theorems for quadratic polynomials

A Shpilka - Proceedings of the 51st Annual ACM SIGACT …, 2019 - dl.acm.org
We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove
that if a finite collection Q, of irreducible polynomials of degree at most 2, satisfy that for …

Radical Sylvester-Gallai Theorem for Tuples of Quadratics

A Garg, R Oliveira, S Peleg… - … Conference (CCC 2023), 2023 - drops.dagstuhl.de
We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic
polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen's theorem …

Strong Algebras and Radical Sylvester-Gallai Configurations

R Oliveira, AK Sengupta - Proceedings of the 56th Annual ACM …, 2024 - dl.acm.org
In this paper, we study the following non-linear generalization of the classical Sylvester-
Gallai configuration. Let K be an algebraically closed field of characteristic 0 and F={F 1,…, F …

Improved hitting set for orbit of roabps

V Bhargava, S Ghosh - Approximation, Randomization, and …, 2021 - drops.dagstuhl.de
The orbit of an n-variate polynomial f (x) over a field 𝔽 is the set {f (Ax+ b)∣ A∈ GL (n, 𝔽)
and b∈ 𝔽ⁿ}, and the orbit of a polynomial class is the union of orbits of all the polynomials in …