# More barriers for rank methods, via a "numeric to symbolic" transfer

## TL;DR

This paper establishes new fundamental limitations in arithmetic complexity, demonstrating that certain rank lower bounds cannot be extended from smaller to larger tensors, and develops algebraic geometry tools for this purpose.

## Contribution

It introduces a novel 'numeric to symbolic' transfer technique that generalizes barrier results in tensor rank methods, with potential applications in complexity theory.

## Key findings

- Proves that small evaluations imply small symbolic rank for tensors.
- Develops a new algebraic geometry tool for transferring relations from numeric to symbolic forms.
- Extends barrier results to higher-order tensors beyond matrices.

## Abstract

We prove new barrier results in arithmetic complexity theory, showing severe limitations of natural lifting (aka escalation) techniques. For example, we prove that even optimal rank lower bounds on $k$-tensors cannot yield non-trivial lower bounds on the rank of $d$-tensors, for any constant $d>k$. This significantly extends recent barrier results on the limits of (matrix) rank methods by Efremenko, Garg, Oliveira and Wigderson, which handles the (very important) case $k=2$. Our generalization requires the development of new technical tools and results in algebraic geometry, which are interesting in their own right and possibly applicable elsewhere. The basic issue they probe is the relation between numeric and symbolic rank of tensors, essential in the proofs of previous and current barriers. Our main technical result implies that for every symbolic $k$-tensor (namely one whose entries are polynomials in some set of variables), if the tensor rank is small for every evaluation of the variables, then it is small symbolically. This statement is obvious for $k=2$. To prove an analogous statement for $k>2$ we develop a "numeric to symbolic" transfer of algebraic relations to algebraic functions, somewhat in the spirit of the implicit function theorem. It applies in the general setting of inclusion of images of polynomial maps, in the form appearing in Raz's elusive functions approach to proving VP $\neq$ VNP. We give a toy application showing how our transfer theorem may be useful in pursuing this approach to prove arithmetic complexity lower bounds.