Minimal Polynomials of Linear Recurrence Sequences

AUTHORS:

• William Stein

sage.matrix.berlekamp_massey.berlekamp_massey(a)

Use the Berlekamp-Massey algorithm to find the minimal polynomial of a linear recurrence sequence $a$.

The minimal polynomial of a linear recurrence $\{a_r\}$ is by definition the unique monic polynomial $g$, such that if $\{a_r\}$ satisfies a linear recurrence $a_{j+k} + b_{j-1} a_{j-1+k} + \cdots + b_0 a_k=0$ (for all $k\geq 0$), then $g$ divides the polynomial $x^j + \sum_{i=0}^{j-1} b_i x^i$.

INPUT:

• a – a list of even length of elements of a field (or domain)

OUTPUT:

the minimal polynomial of the sequence, as a polynomial over the field in which the entries of $a$ live

Warning

The result is only guaranteed to be correct on the full sequence if there exists a linear recurrence of length less than half the length of $a$.

EXAMPLES:

sage: from sage.matrix.berlekamp_massey import berlekamp_massey
sage: berlekamp_massey([1,2,1,2,1,2])
x^2 - 1
sage: berlekamp_massey([GF(7)(1),19,1,19])
x^2 + 6
sage: berlekamp_massey([2,2,1,2,1,191,393,132])
x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673
sage: berlekamp_massey(prime_range(2,38))
x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9