On the existence of non-special divisors of degree g and g-1 in algebraic function fields over Fq

Abstract

We study the existence of non-special divisors of degree g and g − 1 for algebraic function fields of genus g 1 defined over a finite field F q . In particular, we prove that there always exists an effective non-special divisor of degree g 2 if q 3 and that there always exists a non-special divisor of degree g − 1 1 if q 4. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension F q n of F q , when q = 2 r 16.

Determinantal polynomials and the base polynomial of a square matrix over a finite field

“…A theorem proved in Refs. [13,14] and quoted in [15, Proposition 2.2] says that any smooth genus γ curve C such that CðF q Þ ≥ γ þ 1 has a degree γ À 1 line bundle L defined over F q and with h 0 (L) 5 h 1 (L) 5 0. , The lower bound on q in Proposition 4 is not sharp. The existence of a line bundle L as in the proof of Proposition 4 is related to the computational complexity of the multiplication in finite extensions of a finite field ( [13][14][15][16][17]).…”

“…A theorem proved in Refs. [13, 14] and quoted in [15, Proposition 2.2] says that any smooth genus γ curve C such that #C(double-struckFq)≥γ+1 has a degree γ − 1 line bundle L defined over double-struckFq and with h 0 ( L ) = h 1 ( L ) = 0. □…”