Fast range-summable random variables for efficient aggregate estimation
Proceedings of the 2006 ACM SIGMOD international conference on Management of data
The complexity of weighted Boolean #CSP with mixed signs
Theoretical Computer Science
A Complexity Dichotomy for Partition Functions with Mixed Signs
SIAM Journal on Computing
Quantum computing and polynomial equations over the finite field Z2
Quantum Information & Computation
From randomizing polynomials to parallel algorithms
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Algorithms for modular counting of roots of multivariate polynomials
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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We characterize the computational complexity of counting the exact number of satisfying assignments in ($\XOR, \AND$)-formulas in their RSE-representation (i.e., equivalently, polynomials in $GF[2][x_1,\ldots,x_n]$). This problem refrained for some time effords to find a polynomial time solution and the efforts to prove the problem to be $\#P$-complete. Both main results can be generalized to the arbitrary finite fields GF[$q$]. Because counting the number of solutions of polynomials over finite fields is generic for many other algebraic counting problems, the results of this paper settle a border line for the algebraic problems with a polynomial time counting algorithms and for problems which are $\#P$-complete. In \cite{KL89} the couting problem for arbitrary multivariate polynomials over GF[2] has been proved to have randomized polynomial time approximation algorithms.