On polynomial approximations over Z/2kZ

@article{Bhrushundi2017OnPA,
title={On polynomial approximations over Z/2kZ},
author={Abhishek Bhrushundi and Prahladh Harsha and Srikanth Srinivasan},
journal={ArXiv},
year={2017},
volume={abs/1701.06268}
}
• Published 23 January 2017
• Mathematics, Computer Science
• ArXiv
We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)} \pmod {2^k}$. We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with degree $d$ polynomials from $\mathbb{Z}/2^k\mathbb{Z}[x_1,\ldots,x_n]$. Our results are the following…
2 Citations
Towards understanding the approximation of Boolean functions by nonclassical polynomials
The ability of nonclassical polynomials to approximate Boolean functions with respect to both previously studied and new notions of approximation is investigated.
Torus polynomials: an algebraic approach to ACC lower bounds
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2018
It is shown that MAJORITY cannot be approximated by low-degree symmetric torus polynomials, a step towards proving ACC0 lower bounds for the majority function via this approach.

References

SHOWING 1-10 OF 34 REFERENCES
The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic
• Mathematics
• 2011
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function $${f : V \rightarrow \mathbb{C}}$$ on a finite-dimensional
A complex-number Fourier technique for lower bounds on the Mod-m degree
It is shown that any depth-three circuit that computes the Modq function, and consists of an exact threshold gate at the output, Modp gates at the next level, and AND gates of polylog fan-in at the inputs, must be of exponential size.
Lower bounds for approximations by low degree polynomials over Z/sub m/
• Computer Science
Proceedings 16th Annual IEEE Conference on Computational Complexity
• 2001
A Ramsey-theoretic argument is used to obtain the first lower bounds for approximations over Z/sub m/ by nonlinear polynomials by suggesting nonapproximability results imply the first known lower bounds on the top fanin of MAJoMOD/ sub m/oAND/sub O(1)/ circuits.
The distribution of polynomials over finite fields, with applications to the Gowers norms
• Mathematics
Contributions Discret. Math.
• 2009
The main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case it is said that P has small rank.
More Applications of the Polynomial Method to Algorithm Design
• Computer Science, Mathematics
SODA
• 2015
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.
On representations by low-degree polynomials
• R. Smolensky
• Computer Science, Mathematics
Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
• 1993
In the first part of the paper we show that a subset S of a boolean cube B/sub n/ embedded in the projective space P/sup n/ can be approximated by a subset of B/sub n/ defined by nonzeroes of a
Query-efficient algorithms for polynomial interpolation over composites
The interpolation algorithm is used to design algorithms for zero-testing and distributional learning of polynomials over <i>Z</i><inf>m</inf>.
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.
Constant depth circuits, Fourier transform, and learnability
• Computer Science
30th Annual Symposium on Foundations of Computer Science
• 1989
An O(n/sup polylog(/ /sup sup)/ /sup (n)/)-time algorithm for learning functions in AC/sup O/ is obtained and derives a good approximation for the Fourier transform of the function.
New algorithms and lower bounds for circuits with linear threshold gates
An algorithm for evaluating an arbitrary symmetric function of 2no(1) ACC o THR circuits of size 2 no(1), on all possible inputs, in 2n · poly(n) time is given, evidence that non-uniform lower bounds for THR o THR are within reach.