# geneig -- generalized eigenvalue problem

## Synopsis

• Usage:
geneig(kk)
• Inputs:
• kk, a ring, the coefficient ring
• Outputs:
• a list, of the polynomials in the system

## Description

This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 10 solutions found in 1.447 seconds (with a Bezout bound of 243).

Reference: "Homotopy method for general lambda-matrix problems" by M. Chu, T.Y. Li and T. Sauer (pages 528-536).

 i1 : geneig(QQ) 2 2 2 2 2 o1 = {- 10x x + 2x x - x x + x x + 3x x + x x + 2x x + x x + 2x x + 1 6 2 6 3 6 4 6 5 6 1 6 2 6 3 6 4 6 ------------------------------------------------------------------------ 2 2 2 2 x x + 10x + 2x - x + 2x - 2x , 2x x - 11x x + 2x x - 2x x + 5 6 1 2 3 4 5 1 6 2 6 3 6 4 6 ------------------------------------------------------------------------ 2 x x + 2x x + x x + 2x x + x x + 3x x + 2x + 9x + 3x - x - 2x , 5 6 1 6 2 6 3 6 4 6 5 6 1 2 3 4 5 ------------------------------------------------------------------------ 2 2 2 2 2 - x x + 2x x - 12x x - x x + x x + x x + 2x x - 2x x - 2x x - 1 6 2 6 3 6 4 6 5 6 1 6 2 6 4 6 5 6 ------------------------------------------------------------------------ 2 2 2 2 2 x + 3x + 10x + 2x - x , x x - 2x x - x x - 10x x + 2x x + 2x x 1 2 3 4 5 1 6 2 6 3 6 4 6 5 6 1 6 ------------------------------------------------------------------------ 2 + x x - 2x x + 2x x + 3x x + 2x - x + 2x + 12x + x , 3x x + 2 6 3 6 4 6 5 6 1 2 3 4 5 1 6 ------------------------------------------------------------------------ 2 2 2 2 x x + x x + 2x x - 11x x + x x + 3x x - 2x x + 3x x + 3x x - 2 6 3 6 4 6 5 6 1 6 2 6 3 6 4 6 5 6 ------------------------------------------------------------------------ 2x - 2x - x + x + 10x , x + x + x + x + x - 1} 1 2 3 4 5 1 2 3 4 5 o1 : List