Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Complexity of Lattice Problems
Complexity of Lattice Problems
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Learning functions of k relevant variables
Journal of Computer and System Sciences - Special issue: STOC 2003
Computing with polynomials over composites
Computing with polynomials over composites
Computational Complexity: A Conceptual Perspective
Computational Complexity: A Conceptual Perspective
On the Fourier spectrum of symmetric Boolean functions
Combinatorica
On the Minimal Fourier Degree of Symmetric Boolean Functions
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
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We study the following problem raised by von zur Gathen and Roche [GR97]: What is the minimal degree of a nonconstant polynomial f: {0,...,n} → {0,...,m}? Clearly, when m = n the function f(x) = x has degree 1. We prove that when m = n - 1 (i.e. the point {n} is not in the range), it must be the case that deg(f) = n - o(n). This shows an interesting threshold phenomenon. In fact, the same bound on the degree holds even when the image of the polynomial is any (strict) subset of {0,...,n}. Going back to the case m = n, as we noted the function f(x) = x is possible, however, we show that if one excludes all degree 1 polynomials then it must be the case that deg(f) = n - o(n). Moreover, the same conclusion holds even if m = O(n1.475--ε). In other words, there are no polynomials of intermediate degrees that map {0,...,n} to {0,...,m}. Furthermore, we give a meaningful answer when m is a large polynomial, or even exponential, in n. Roughly, we show that if m n/c d), for some constant c, then either deg(f) ≤ d - 1 (e.g. f(x) = (x-n/2d - 1) is possible) or deg(f) ≥ n/3 - O(d log n). So, again, no polynomial of intermediate degree exists for such m. We achieve this result by studying a discrete version of the problem of giving a lower bound on the minimal L∞, norm that a monic polynomial of degree d obtains on the interval [-1,1]. We complement these results by showing that for every d = o(√n/log n) there exists a polynomial f: {0,...,n} → {0,...,O(nd+0.5)} of degree n/3 - O(d log n) ≤ deg(f) ≤ n - O(d log (n)). Our proofs use a variety of techniques that we believe will find other applications as well. One technique shows how to handle a certain set of diophantine equations by working modulo a well chosen set of primes (i.e. a Boolean cube of primes). Another technique shows how to use lattice theory and Minkowski's theorem to prove the existence of a polynomial with certain properties.