Polynomials Modulo Composite Numbers: Ax-Katz type theorems for the structure of their solution sets (original) (raw)

On the reduction in multiplicative complexity achieved by the polynomial residue number system

IEEE Transactions on Signal Processing, 1992

The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.

The complexity of the Chinese Remainder Theorem

arXiv (Cornell University), 2023

The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class V generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless V is the class of semilattices, the problem is either coNP-complete or tractable.

The Polynomial Method in Circuit Complexity Applied to Algorithm Design (Invited Talk)

In circuit complexity, the polynomial method is a general approach to proving circuit lower bounds in restricted settings. One shows that functions computed by sufficiently restricted circuits are "correlated" in some way with a low-complexity polynomial, where complexity may be measured by the degree of the polynomial or the number of monomials. Then, results limiting the capabilities of low-complexity polynomials are extended to the restricted circuits. Old theorems proved by this method have recently found interesting applications to the design of algorithms for basic problems in the theory of computing. This paper surveys some of these applications, and gives a few new ones.

Complex polynomials and circuit lower bounds for modular counting

Computational Complexity, 1994

We study the power of constant-depth circuits containing negation gates, unbounded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of size O(n) (where n is the number of inputs) containing O(n) MAJORITY gates can determine whether the sum of the input bits is divisible by k, for any fixed k > 1~ whereas it is known that this requires exponentialsize circuits if we have no MAJORITY gates. Our main result is that a constant-depth circuit of size 2 n~ containing n ~ MAJORITY gates cannot determine if the sum of the input bits is divisible by k; moreover, such a circuit must give the wrong answer on a constant fraction of the inputs. This result was previously known only for k = 2. We prove this by obtaining an approximate representation of the behavior of constant-depth circuits by multivariate complex polynomials.

On Threshold Circuits and Polynomial Computation

SIAM Journal on Computing, 1992

A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field Z P (n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P (n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) ≥ n and depth D(n) can also be computed by Z P (n) Circuits of size O(S(n) log S(n) + nP (n) log P (n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by Z P (n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O(1 2 (S(n) log P (n)) 1+) and depth O(1 5 D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O(1) and depth O(1). More generally, any analytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2 −n c , for any constant c, by Threshold Circuits of polynomial size and constant depth. In addition, integer and polynomial division, FFT, polynomial interpolation, Chinese Remaindering, all the elementary symmetric functions, banded matrix inverse, and triangular Toeplitz matrix inverse can be exactly computed by Threshold Circuits of polynomial size and constant depth. All these results and simulations hold for polytime uniform circuits. This paper also gives a corresponding simulation of logspace uniform Z P (n) Circuits by logspace uniform Threshold Circuits requiring an additional multiplying factor of O(log log log P (n)) depth. Finally, purely algebraic methods for lower bounds for Z P (n) Circuits are developed. Using degree arguments, a Depth Hierarchy Theorem for Z P (n) Circuits is proved: for any S(n) ≥ n, D(n) = O(S(n) c) for some constant c < 1, and prime P (n) where 6(S(n)/D(n)) D(n) < P (n) ≤ 2 n , there exists explicitly constructible functions computable by Z P (n) Circuits of size S(n) and depth D(n), but provably not computable by Z P (n) Circuits of size S(n) c and depth o(D(n)) for any constant c ≥ 1.

Representing Boolean functions as polynomials modulo composite numbers

Computational Complexity, 1994

De ne the MOD m -degree of a boolean function F to be the smallest degree of any polynomial P, over the ring of integers modulo m, such that for all 0-1 assignmentsx, F(x) = 0 i P(x) = 0. We obtain the unexpected result that the MOD m -degree of the OR of N variables is O( r p N), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MOD m -degree of both the MOD n and :MOD n functions is N (1) exactly when there is a prime dividing n but not m. The MOD m -degree of the MOD m function is 1; we show that the MOD m -degree of :MOD m is N (1) if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MOD m P classes (such as P) have this structure: MOD m P is closed under complementation and union i m is a prime power, and MOD n P is a subset of MOD m P i all primes dividing n also divide m.

On the complexity of quantum ACC

Proceedings 15th Annual IEEE Conference on Computational Complexity, 2000

For any q > 1, let MOD q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q, t > 1, MOD q is equivalent to MOD t (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC (0) , ACC[q], and ACC, denoted QAC (0) wf , QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, proving that QAC

Computational complexity of some problems involving congruences on algebras

Theoretical Computer Science, 2002

We prove that several problems concerning congruences on algebras are complete for nondeterministic log-space. These problems are: determining the congruence on a given algebra generated by a set of pairs, and determining whether a given algebra is simple or subdirectly irreducible. We also consider the problem of determining the smallest fully invariant congruence on a given algebra containing a given set of pairs. We prove that this problem is complete for nondeterministic polynomial time.

The Hardness of Polynomial Equation Solving

Foundations of Computational Mathematics, 2003

Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.