All reports by Author Rafael Mendes de Oliveira:

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TR16-122
| 11th August 2016
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Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf#### Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound

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TR14-157
| 27th November 2014
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Rafael Mendes de Oliveira, Amir Shpilka, Ben Lee Volk#### Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

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TR14-056
| 18th April 2014
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Zeev Dvir, Rafael Mendes de Oliveira#### Factors of Sparse Polynomials are Sparse

Revisions: 1
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Comments: 1

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TR14-003
| 10th January 2014
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Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka#### Testing Equivalence of Polynomials under Shifts

Revisions: 2
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Comments: 1

Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf

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 ...
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Rafael Mendes de Oliveira, Amir Shpilka, Ben Lee Volk

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 >>>

Zeev Dvir, Rafael Mendes de Oliveira

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 ...
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Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka

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 >>>