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Electronic Colloquium on Computational Complexity

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All reports by Author Rafael Mendes de Oliveira:

TR22-131 | 18th September 2022
Rafael Mendes de Oliveira, Akash Sengupta

Radical Sylvester-Gallai for Cubics

Let $\mathcal{F} = \{F_1, \ldots, F_m\}$ be a finite set of irreducible homogeneous multivariate polynomials of degree at most $3$ such that $F_i$ does not divide $F_j$ for $i\neq j$.
We say that $\mathcal{F}$ is a cubic radical Sylvester-Gallai configuration if for any two distinct $F_i,F_j$ there exists a ... more >>>

TR22-037 | 10th March 2022
Abhibhav Garg, Rafael Mendes de Oliveira, Akash Sengupta

Robust Radical Sylvester-Gallai Theorem for Quadratics

We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials, generalizing the result in [S'20].
More precisely, given a parameter $0 < \delta \leq 1$ and a finite collection $\mathcal{F}$ of irreducible and pairwise independent polynomials of degree at most 2, we say that $\mathcal{F}$ is a ... more >>>

TR19-140 | 7th October 2019
Ankit Garg, Visu Makam, Rafael Mendes de Oliveira, Avi Wigderson

Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings.

Revisions: 1

We consider the problem of outputting succinct encodings of lists of generators for invariant rings. Mulmuley conjectured that there are always polynomial sized such encodings for all invariant rings. We provide simple examples that disprove this conjecture (under standard complexity assumptions).

more >>>

TR19-019 | 19th February 2019
Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi

Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\mathop{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas (MFCS ... more >>>

TR17-162 | 26th October 2017
Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, Avi Wigderson

Barriers for Rank Methods in Arithmetic Complexity

Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than ... more >>>

TR16-122 | 11th August 2016
Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf

Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound

One of the most important open problems in the theory
of error-correcting codes is to determine the
tradeoff between the rate $R$ and minimum distance $\delta$ of a binary
code. The best known tradeoff is the Gilbert-Varshamov bound,
and says that for every $\delta \in (0, 1/2)$, there are ... more >>>

TR14-157 | 27th November 2014
Rafael Mendes de Oliveira, Amir Shpilka, Ben Lee Volk

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models.

For depth-3 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 2\delta/3}))$. ... more >>>

TR14-056 | 18th April 2014
Zeev Dvir, Rafael Mendes de Oliveira

Factors of Sparse Polynomials are Sparse

Revisions: 1 , Comments: 1

We show that if $f(x_1,\ldots,x_n)$ is a polynomial with $s$ monomials and $g(x_1,\ldots,x_n)$ divides $f$ then $g$
has at most $\max(s^{O(\log s \log\log s)},d^{O(\log d)})$ monomials, where $d$ is a bound on the individual degrees
of $f$. This answers a question of von zur Gathen and Kaltofen (JCSS ... more >>>

TR14-003 | 10th January 2014
Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka

Testing Equivalence of Polynomials under Shifts

Revisions: 2 , Comments: 1

Two polynomials $f, g \in F[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in {F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our ... more >>>

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