On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Efficient parallel evaluation of straight-line code and arithmetic circuits
SIAM Journal on Computing
Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Efficient Gaussian Elimination Method for Symbolic Determinants and Linear Systems
ACM Transactions on Mathematical Software (TOMS)
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic
FCT '85 Fundamentals of Computation Theory
Optimal Parallel Evaluation of Tree-Structured Computations by Raking
AWOC '88 Proceedings of the 3rd Aegean Workshop on Computing: VLSI Algorithms and Architectures
Determinantal formula for the chow form of a toric surface
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Modern Computer Algebra
Sylvester-resultants for bivariate polynomials with planar newton polygons
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
On the complexity of computing determinants
Computational Complexity
Characterizing Valiant's algebraic complexity classes
Journal of Complexity
The multivariate resultant is NP-hard in any characteristic
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Journal of Symbolic Computation
Arithmetic circuits: The chasm at depth four gets wider
Theoretical Computer Science
On the complexity of the multivariate resultant
Journal of Complexity
Sparse multivariate function recovery from values with noise and outlier errors
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits
Computational Complexity
On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation
Theory of Computing Systems
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Suppose the polynomials f and g in K[x1,...,xr] over the field K are determinants of non-singular m x m and n x n matrices, respectively, whose entries are in K ∪ x1,...,xr. Furthermore, suppose h = f/g is a polynomial in K[x1,..., xr]. We construct an s x s matrix C whose entries are in K ∪ x1,...,xr, such that h = det(C) and s = γ (m+n)6, where γ = O(1) if K is an infinite field or if for the finite field K = F{q} with q elements we have m = O(q), and where γ = (logq m)1+o(1) if q = o(m). Our construction utilizes the notion of skew circuits by Toda and WSK circuits by Malod and Portier. Our problem was motivated by resultant formulas derived from Chow forms. Additionally, we show that divisions can be removed from formulas that compute polynomials in the input variables over a sufficiently large field within polynomial formula size growth.