Rational functions with partial quotients of small degree in their continued fraction expansion

Abstract

For a rational functionf/g=f(x)/g(x) over a fieldF with ged (f,g)=1 and deg (g)≥1 letK(f/g) be the maximum degree of the partial quotients in the continued fraction expansion off/. ForfεF[x] with deg (f)=k≥1 andf(O)≠O putL(f)=K(f(x)/x ^k). It is shown by an explicit construction that for every integerb with 1≤b≤k there exists anf withL(f)=b. IfF=F ₂, the binary field, then for everyk there is exactly onefεF ₂[x] with deg (f)=k,f(O)≠O, andL(f)=1. IfF _q is the finite field withq elements andgεF _q [x] is monic of degreek≥1, then there exists a monic irreduciblefεF _q [x] with deg (f)=k, gcd (f,g)=1, andK(f/g)<2+2 (logk)/logq, where the caseq=k=2 andg(x)=x ²+x+1 is excluded. An analogous existence theorem is also shown for primitive polynomials over finite fields. These results have applications to pseudorandom number generation.