Algebraic hardness versus randomness in low characteristic

@article{Andrews2020AlgebraicHV,
  title={Algebraic hardness versus randomness in low characteristic},
  author={Robert Andrews},
  journal={Proceedings of the 35th Computational Complexity Conference},
  year={2020}
}
  • Robert Andrews
  • Published 21 May 2020
  • Computer Science, Mathematics
  • Proceedings of the 35th Computational Complexity Conference
We show that lower bounds for explicit constant-variate polynomials over fields of characteristic p > 0 are sufficient to derandomize polynomial identity testing over fields of characteristic p. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no… Expand
2 Citations
Demystifying the border of depth-3 algebraic circuits
Border complexity of polynomials plays an integral role in GCT (Geometric complexity theory) approach to P 6= NP. It tries to formalize the notion of ‘approximating a polynomial’ via limitsExpand
Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits
TLDR
A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧. Expand

References

SHOWING 1-10 OF 48 REFERENCES
Hardness vs Randomness for Bounded Depth Arithmetic Circuits
TLDR
It is shown that if there is a family of explicit polynomials {fn}, where fn is of degree O(log2 n/log2 logn) in n variables such that fn cannot be computed by a depth Δ arithmetic circuits of size poly(n), thenthere is a deterministic sub-exponential time algorithm for polynomial identity testing of arithmetic circuit of depth Δ − 5. Expand
Hardness-randomness tradeoffs for bounded depth arithmetic circuits
TLDR
The methods of Impagliazzo and Kabanets imply that if the authors can derandomize polynomial identity testing for bounded depth circuits then NEXP does not have bounded depth arithmetic circuits, that is, either NEXP ⊄ P/poly or the Permanent is not computable byPolynomial size bounded Depth d arithmetic circuits. Expand
Bootstrapping variables in algebraic circuits
TLDR
The idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially wrt variables, and implies a lower bound that is a bit stronger than Kabanets-Impagliazzo (STOC 2003). Expand
Derandomization from Algebraic Hardness: Treading the Borders
TLDR
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. Expand
Hardness vs Randomness
TLDR
A new construction of a pseudorandom bit generator that stretches a short string of truly random bits into a long string that looks random to any algorithm from a complexity class C using an arbitrary function that is hard for C is presented. Expand
Hardness-Randomness Tradeoffs for Algebraic Computation
TLDR
This survey discusses some of the classical results, as well as some recent ones, that establish a close connection between the question of proving algebraic circuits lower bounds and that of derandomizing polynomial identity testing. Expand
Arithmetic Circuits: A survey of recent results and open questions
TLDR
The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades. Expand
Progress on Polynomial Identity Testing
TLDR
Nitin Saxena gives in this survey a beautiful overview of several recent results dealing with the complexity of Polynomial Identity Testing. Expand
Primality and identity testing via Chinese remaindering
  • Manindra Agrawal, S. Biswas
  • Mathematics, Computer Science
  • 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
  • 1999
TLDR
These algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods, e.g. the Schwartz-Zippel test, the Chen-Kao (1997) test and the Lewin-Vadhan test. Expand
Arithmetic Complexity in Ring Extensions
TLDR
It is obtained that the elementary symmetric polynomials have formulas of size n O(log log n) over any field, and that division gates can be efficiently eliminated from circuits. Expand
...
1
2
3
4
5
...