Results 181 to 190 of about 6,241 (212)

On Multiplication of Polynomials Modulo a Polynomial

SIAM Journal on Computing, 1980
The multiplicative complexity of the direct product of algebras $A_p $ of polynomials modulo a polynomial P is studied. In particular, we show that if P and Q are irreducible polynomials then the multiplicative complexity of $A_{\text{P}} \times A_{\text{Q}} $ is $2\deg ({\text{P}})\deg ({\text{Q}}) - {\text{k}}$, where k is the number of factors of P ...
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Sparse multiplication for skew polynomials

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, 2020
Consider the skew polynomial ring L[x; σ], where L is a field and σ is an automorphism of L of order r. We present two randomized algorithms for the multiplication of sparse skew polynomials in L[x;σ]. The first algorithm is Las Vegas; it relies on evaluation and interpolation on a normal basis, at successive powers of a normal element. For inputs A, B
Mark Giesbrecht, Qiao-Long Huang, Éric Schost   +2 more
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Bivariate polynomial multiplication

Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), 2002
We study the multiplicative complexity and the rank of the multiplication in the local algebras R/sub m,n/=k[x,y]/(x/sup m+1/,y/sup n+1/) and T/sub n/=k[x,y]/(x/sup n+1/,x/sup n/y,...,y/sup n+1/) of bivariate polynomials. We obtain the lower bounds (21/3-0(1))/spl middot/dim R/sub m,n/, and (2 1/2 -0(1))/spl middot/dim T/sub n/ for the multiplicative ...
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New Algorithms for Polynomial Multiplication

SIAM Journal on Computing, 1979
Exploiting the structure of the 2-dimensional sorting problem associated with the polynomial product has been the strategy in the design of certain algorithms which are faster for a large class of problems than those found in the literature. First a parallel is drawn between GEN–MULT and Horowitz’s SORT–MULT algorithm [A sorting algorithm for ...
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Multiplication of a Schubert polynomial by a Schur polynomial

Annals of Combinatorics, 1997
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Multiplication operator on a matrix polynomial

Ukrainian Mathematical Journal, 1992
Summary: It is shown, how the study of the perturbed multiplication operator by a matrix polynomial in the space \(L_ 2(\mathbb{R},\mathbb{C}^ n)\) can be reduced to the study of the perturbed multiplication operator by the independent variable in the space \(L_ 2(\mathbb{R},\omega,\mathbb{C}^ n)\) with weight \(\omega\), fulfilling the Muckenhoupt ...
Mikityuk, Ya. V., Al'-Tundzhi, M.
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On Integer Polynomials with Multiple Roots

Mathematika, 2007
The authors prove the following result Theorem 1. Let \(P(X)\) be a polynomial of degree \(n \geq 2\). Denote by \(\alpha\) a zero of \(P\) of order \(s\) and \(\beta\) another zero of \(P\) of order \(t\geq s\), then \[ | \alpha-\beta| \geq 2^{-n/(2t)}(n+1)^{-n/(2s)-3n/(4st)}\text{H}(P)^{-n/(st)}\max\{1,| \alpha| \}\max\{1,| \beta| \}, \] where ...
Amou, Masaaki, Bugeaud, Yann
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Code Generation for Polynomial Multiplication

2009
We discuss the family of "divide-and-conquer" algorithms for polynomial multiplication, that generalize Karatsuba's algorithm. We give explicit versions of transposed and short products for this family of algorithms and describe code generation techniques that result in high-performance implementations.
Ling Ding, Éric Schost
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POLYNOMIALS WITH MULTIPLE ROOTS AT 1

International Journal of Number Theory, 2014
We prove that every polynomial P(x) = 1 + a1x + ⋯ + adxd∈ ℂ[x] has a root at x = 1 of order at most [Formula: see text], where ε > 0, provided that the quantities H*(P) = max1 ≤ j ≤ d|aj| and d/ log H*(P) are sufficiently large (the order of the root of P at x = 1 is zero if P(1) ≠ 0).
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Bivariate polynomial multiplication patterns

1995
Motivated by multiplication of numerical univariate polynomials with various kinds of truncation we study corresponding bivariate problems A(x, y)·B(x, y) = C(x, y) in the algebraic setting with indeterminate coefficients over suitable ground fields, counting essential multiplications only.
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