Abstract
$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\Image}{{\mathrm{Image}}}$ It is a basic fact of linear algebra that the image of the curve $f(x)=(x^1,x^2,x^3,\ldots,x^m)$, say over $\C$, is not contained in any $(m-1)$-dimensional affine subspace of $\C^m$. In other words, the image of $f$ is not contained in the image of any polynomial mapping $\Gamma:\C^{m-1} \rightarrow \C^m$ of degree 1 (that is, an affine mapping). Can one give an explicit example of a polynomial curve $f:\C \rightarrow \C^m$ such that the image of $f$ is not contained in the image of any polynomial mapping $\Gamma:\C^{m-1} \rightarrow \C^m$ of degree $2$ ? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit $f$ as above (with the right notion of explicitness), of degree up to $2^{m^{o(1)}}$, implies super-polynomial lower bounds for computing the permanent over$~\C$. More generally, we say that a polynomial mapping $f:\F^{n} \rightarrow \F^m$ is $(s,r)$-elusive, if for every polynomial mapping $\Gamma:\F^{s} \rightarrow \F^m$ of degree $r$, $\Image(f) \not \subset \Image(\Gamma)$. We show that for many settings of the parameters $n,m,s,r$, explicit constructions of elusive polynomial mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every $r < \log n$, we give an explicit example of a polynomial mapping $f:\F^{n} \rightarrow \F^{n^2}$, of degree $O(r)$, that is $(s,r)$-elusive for $s = n^{1+\Omega(1/r)}$. We use this to construct for any $r$, an explicit example of an $n$-variate polynomial of total-degree $O(r)$, with coefficients in $\{0,1\}$, such that any depth-$r$ arithmetic circuit for this polynomial (over any field) is of size $\geq n^{1+\Omega(1/r)}$. In particular, for any constant $r$, this gives a constant degree polynomial such that any depth $r$ arithmetic circuit for this polynomial is of size $\geq n^{1+\Omega(1)}$. Previously, only lower bounds of the type $\Omega(n \cdot \lambda_r (n))$, where the $\lambda_r$ are extremely slowly growing functions, were known for constant-depth arithmetic circuits for polynomials of constant degree (actually, for linear functions).
Highlights
We present a family of problems that are very simple to describe, that seem natural to study from several different points of view, that are seemingly unrelated to arithmetic circuit complexity, and whose solution would give strong lower bounds for the size of general arithmetic circuits
We show that if one can give a polynomial-time Turing machine that on input Γ, as above, outputs an explicit polynomial mapping f : Fn → Fm of degree at most poly(n), such that Image( f ) ⊂ Image(Γ), one obtains an explicit lower bound of Ω(n10) for the size of arithmetic circuits
One can obtain “win-win” results, such as: either the problem of finding a point outside the image of a polynomial mapping Γ is hard, in which case we have an example of a hard problem, or, otherwise, there exists an explicit lower bound of Ω(n10) for the size of arithmetic circuits
Summary
We present a family of problems that are very simple to describe, that seem natural to study from several different points of view (such as, geometric, algebraic and combinatorial), that are seemingly unrelated to arithmetic circuit complexity, and whose solution would give strong (up to exponential) lower bounds for the size of general arithmetic circuits. If there exists an explicit (s, r)-elusive polynomial mapping, f : Fn → Fm (of degree at most poly(n)), any arithmetic circuit for the permanent, over F, is of super-polynomial size. We show that if one can give a polynomial-time Turing machine that on input Γ, as above, outputs an explicit polynomial mapping f : Fn → Fm of degree at most poly(n), such that Image( f ) ⊂ Image(Γ), one obtains an explicit lower bound of Ω(n10) for the size of arithmetic circuits. One can obtain “win-win” results, such as: either the problem of finding a point outside the image of a polynomial mapping Γ is hard (when Γ is given as an input), in which case we have an example of a hard problem, or, otherwise, there exists an explicit lower bound of Ω(n10) for the size of arithmetic circuits (or both). Only slightly super-linear lower bounds were known for constant-depth arithmetic circuits, for polynomials of constant degree (for linear functions)
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