Uniform closure properties of P-computable functions

Val iant [1] introduced the notion of a family of 19computable polynomials as those mul t ivar ia te polynomials of polynomial ly-bounded degree and straight-l ine computat ion length. He raised the question of whether pcomputable families would be closed under natural mathemat ica l operations and showed tha t this is true for taking repeated part ial derivatives in a single variable, whereas by taking repeated part ial derivatives in many variables one can obtain the general permanent from a polynomial-sized formula.

[1]  Erich Kaltofen,et al.  Polynomial-Time Reductions from Multivariate to Bi- and Univariate Integral Polynomial Factorization , 1985, SIAM J. Comput..

[2]  Erich Kaltofen,et al.  Greatest common divisors of polynomials given by straight-line programs , 1988, JACM.

[3]  Erich Kaltofen.  Computing with polynomials given by straight-line programs I: greatest common divisors , 1985, STOC '85.

[4]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[5]  J. von zur Gathen.  Factoring sparse multivariate polynomials , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[6]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

[7]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[8]  Erich Kaltofen,et al.  Effective Hilbert Irreducibility , 1984, Inf. Control..

[9]  C.P. Schnorr.  Improved Lower Bounds on the Number of Multiplications/Divisions which are Necessary of Evaluate Polynomials , 1978, Theor. Comput. Sci..

[10]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[11]  Leslie G. Valiant,et al.  Fast Parallel Computation of Polynomials Using Few Processors , 1983, SIAM J. Comput..

[12]  Arnold Schnhage.  Schnelle Multiplikation von Polynomen ber Krpern der Charakteristik 2 , 1977 .

[13]  Laurent Hyafil,et al.  On the parallel evaluation of multivariate polynomials , 1978, SIAM J. Comput..

[14]  Keith O. Geddes,et al.  A Comparison of Algorithms for the Symbolic Computation of Padé Approximants , 1984, EUROSAM.

[15]  Peter J. Weinberger.  Finding the Number of Factors of a Polynomial , 1984, J. Algorithms.

[16]  KaltofenErich.  Greatest common divisors of polynomials given by straight-line programs , 1988 .

[17]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[18]  Joachim von zur Gathen.  Parallel algorithms for algebraic problems , 1983, STOC '83.

[19]  Robert T. Moenck,et al.  Fast computation of GCDs , 1973, STOC.

[20]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[21]  J. Shepherdson,et al.  Effective procedures in field theory , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[22]  Joachim von zur Gathen,et al.  Irreducibility of Multivariate Polynomials , 1985, J. Comput. Syst. Sci..

[23]  Richard J. Lipton,et al.  Evaluation of polynomials with super-preconditioning , 1976, STOC '76.

[24]  Claus-Peter Schnorr.  Improved Lower Bounds on the Number of Multiplications/Divisions Which Are Necessary to Evaluate Polynomials , 1977, MFCS.

[25]  Erich Kaltofen,et al.  Computing with polynomials given by straight-line programs II sparse factorization , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[26]  Peter J. Weinberger,et al.  Factoring Polynomials Over Algebraic Number Fields , 1976, TOMS.

[27]  Joachim von zur Gathen,et al.  Parallel algorithms for algebraic problems , 1983, SIAM J. Comput..

[28]  Erich Kaltofen,et al.  Factoring Sparse Multivariate Polynomials , 1983, J. Comput. Syst. Sci..