There are lots of algorithms for factoring in Z[x]. But what about power series with integer coefficients? Is there a good algorithm? Knuth Vol 2 is silent. Any hints would be appreciated! There are two versions, depending on whether you work to order n, or with the full power series. Version 1: Given 2 3 4 5 6 7 8 t3 := 1 + 3 x + 7 x + 17 x + 42 x + 113 x + 321 x + 976 x + 3109 x + 9 10 11 12 10258 x + 34698 x + 119554 x + O(x ), find t1 and t2 such that t1*t2 = t3 (mod x^12). Answer:
t1; 2 3 4 5 6 7 8 9 1 + x + 2 x + 5 x + 14 x + 42 x + 132 x + 429 x + 1430 x + 4862 x
10 11 + 16796 x + 58786 x + ...
t2; 2 3 4 5 6 7 8 9 10 1 + 2 x + 3 x + 5 x + 7 x + 11 x + 13 x + 17 x + 19 x + 23 x + 29 x
11 + 31 x + ... Do any of Maple, Mma, Maxima, PARI, Magma, etc. have a command for factoring in this ring? Version 2 (harder): Factor: 1 + 2*Sum_{m >= 1} x^(m^2) = 1 + 2x + 2x^4 + 2x^9 + 2x^16 + ... Answer: Product_{n >= 1} (1-x^2n)(1+x^(2n-1))^2 Neil Sloane