9 research outputs found
Sub-quadratic time for Riemann-Roch spaces. The case of smooth divisors over nodal plane projective curves
International audienceWe revisit the seminal Brill-Noether algorithm in the rather generic situation of smooth divisors over a nodal plane projective curve. Our approach takes advantage of fast algorithms for polynomials and structured matrices. We reach sub-quadratic time for computing a basis of a Riemann-Roch space. This improves upon previously known complexity bounds
Beating binary powering for polynomial matrices
The th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in . When Fast Fourier Transform (FFT) is available, the resulting arithmetic complexity is \emph{softly linear} in , i.e. linear in with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is \emph{purely linear} in , even in absence of FFT. The key result making this improvement possible is that the entries of the th power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of . Similar algorithms are proposed for two related problems: computing the th term of a C-recursive sequence of polynomials, and modular exponentiation to the power for bivariate polynomials
On the factorization of polynomials over algebraic fields
SIGLEAvailable from British Library Document Supply Centre- DSC:DX86869 / BLDSC - British Library Document Supply CentreGBUnited Kingdo