# HurwitzMatrix -- a specified principle submatrix of the Hurwitz matrix of a univariate polynomial

## Synopsis

• Usage:
HurwitzMatrix(f,k)
• Inputs:
• f, , a rational univariate polynomial of degree n
• k, an integer, a nonnegative integer that determines the dimensions of a square submatrix of the $n\times n$ Hurwitz matrix of f.
• Outputs:
• , the $k\times k$ submatrix $H_{k}$ generated by the corresponding leading principal minor of the $n\times n$ Hurwitz matrix $H$ of f.

## Description

This computes the $k\times k$ submatrix $H_{k}$ of the corresponding leading principal minor of the $n\times n$ Hurwitz matrix $H$ of a rational univariate polynomial f of degree n with positive leading coefficient and degree at least 1. The polynomial, however, is not necessarily from a univariate polynomial ring.

 i1 : R = QQ[x] o1 = R o1 : PolynomialRing i2 : f = 3*x^4 - 7*x^3 + 5*x - 7 4 3 o2 = 3x - 7x + 5x - 7 o2 : R i3 : HurwitzMatrix(f) o3 = | -7 5 0 0 | | 3 0 -7 0 | | 0 -7 5 0 | | 0 3 0 -7 | 4 4 o3 : Matrix QQ <--- QQ i4 : HurwitzMatrix(f,4) o4 = | -7 5 0 0 | | 3 0 -7 0 | | 0 -7 5 0 | | 0 3 0 -7 | 4 4 o4 : Matrix QQ <--- QQ i5 : HurwitzMatrix(f,3) o5 = | -7 5 0 | | 3 0 -7 | | 0 -7 5 | 3 3 o5 : Matrix QQ <--- QQ

We can also use mutliple variables to represent unknown coefficients. Note that we create another ring S so that x and y are not considered variables in the same ring and so confuse the monomials $x$ or $y$ with $xy$.

 i6 : S = R[y] o6 = S o6 : PolynomialRing i7 : g = y^3 + 2*y^2 + y - x + 1 3 2 o7 = y + 2y + y - x + 1 o7 : S i8 : HurwitzMatrix(g,3) o8 = | 2 -x+1 0 | | 1 1 0 | | 0 2 -x+1 | 3 3 o8 : Matrix R <--- R i9 : HurwitzMatrix(g,2) o9 = | 2 -x+1 | | 1 1 | 2 2 o9 : Matrix R <--- R i10 : HurwitzMatrix(g,1) o10 = | 2 | 1 1 o10 : Matrix R <--- R