Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Derandomization from Algebraic Hardness   A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to
\mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough
circuit size and degree, if $C$ is nonzero, then $C\circ G$ is nonzero. In this
paper, we give a new construction of such an HSG assuming that we have an
explicit polynomial of sufficient hardness. Formally, we prove the following
over any field of characteristic zero:
  Let $k\in \mathbb{N}$ and $\delta > 0$ be arbitrary constants. Suppose
$\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials
such that $\operatorname{deg} P_d = d$ and $P_d$ requires algebraic circuits of
size $d^\delta$. Then, there are explicit hitting sets of polynomial size for
$\mathsf{VP}$.
  This is the first HSG in the algebraic setting that yields a complete
derandomization of polynomial identity testing (PIT) for general circuits from
a suitable algebraic hardness assumption. As a direct consequence, we show that
even saving a single point from the "trivial" explicit, exponential sized
hitting sets for constant-variate polynomials of low individual degree which
are computable by small circuits, implies a deterministic polynomial time
algorithm for PIT. More precisely, we show the following:
  Let $k\in \mathbb{N}$ and $\delta > 0$ be arbitrary constants. Suppose for
every $s$ large enough, there is an explicit hitting set of size at most
$((s+1)^k - 1)$ for the class of $k$-variate polynomials of individual degree
$s$ that are computable by size $s^\delta$ circuits. Then there is an explicit
hitting set of size $\operatorname{poly}(s)$ for the class of $s$-variate
polynomials, of degree $s$, that are computable by size $s$ circuits.
  As a consequence, we give a deterministic polynomial time construction of
hitting sets for algebraic circuits, if a strengthening of the
$\tau$-Conjecture of Shub and Smale is true.

0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
18
The categories of this paper are: econ
2
The categories of this paper are: math
14
The categories of this paper are: math-ph