2013/09/08 by Vesselin Dimitrov, Dimitrov, Vesselin
Computer Science · Engineering · Mathematics · #11G30 #11G50 #30C80 #30C85 #30F15 #41A20 #41A21 #41A58 #Coding theory and cryptography #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1309.1920
openalex publication_date 2013/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a generalization of the "Hadamard quotient theorem" of Pourchet and van der Poorten. A particular case of our conjecture states that if f := ∑n ≥ 0 a(n)xn and g := ∑n ≥ 0 b(n)xn represent, respectively, an algebraic and a rational function over a global field K such that b(n) ≠ 0 for all n and the coefficients of the power series h := ∑n ≥ 0 a(n)/b(n)xn are contained in a finitely generated ring, then h is algebraic. We prove this conjecture if either (i) g has a simple pole of a strictly maximal absolute value at some place; or (ii) or poles of g are simple, there is a positive density δ> 0 of places which split completely in the field generated by the poles of g and at which all b(n) are units, and with d := [K(t,f):K(f)], the local radii of convergence Rv of h at the places v of K satisfy ∑v log+Rv-1 ≤ δ/12d4.