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TR20-039 | 25th March 2020 20:57
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#### Lower bounds on the sum of 25th-powers of univariates lead to complete derandomization of PIT

TR20-039
Authors:

Pranjal Dutta,

Nitin Saxena,

Thomas Thierauf
Publication: 25th March 2020 23:21

Downloads: 472

Keywords:

#P/poly,

CH,

circuit,

Hitting Set,

lower bounds,

Matrix Rigidity,

monomial,

PIT,

powers,

squares,

support,

VNP,

VP
**Abstract:**
We consider the univariate polynomial $f_d:=(x+1)^d$ when represented as a sum of constant-powers of univariate polynomials. We define a natural measure for the model, the support-union, and conjecture that it is $\Omega(d)$ for $f_d$.

We show a stunning connection of the conjecture to the two main problems in algebraic complexity: Polynomial Identity Testing (PIT) and VP vs VNP. Our conjecture on $f_d$ implies blackbox-PIT in P. Assuming the Generalized Riemann Hypothesis (GRH), it also implies VP $\neq$ VNP. No such connection to PIT, from lower bounds on constant-powers representation of polynomials was known before. We establish that studying the expression of $(x+1)^d$, as the sum of $25^{th}$-powers of univariates, suffices to solve the two major open questions.

In support, we show that our conjecture holds over the integer ring of any number field. We also establish a connection with the well-studied notion of matrix rigidity.