# bivariateDiagEntries -- computes diagonal entries and eigenvalues for a determinantal representation of a bivariate polynomial

## Synopsis

• Usage:
bivariateDiagEntries f
bivariateDiagEntries(f, Tolerance => 1e-6)
• Inputs:
• f, , a bivariate polynomial with real coefficients
• Optional inputs:
• Tolerance => ..., default value .000001
• Outputs:
• , of eigenvalues and diagonal entries of a determinantal representation of $f$

## Description

This method computes the eigenvalues and diagonal entries of a monic symmetric determinantal representation of a real bivariate polynomial $f$, or gives an error if certain necessary conditions for existence of such a representation are not met. For a symmetric determinantal representation $f = det(I + x_1A_1 + x_2A_2)$, this method computes diagonal entries and eigenvalues of $A_1$ and $A_2$. The output is a 4-tuple of column vectors: (eigenvalues of A_1, eigenvalues of $A_2$, diagonal entries of $A_1$, diagonal entries of $A_2$).

When working over an InexactFieldFamily like RR or CC, the option Tolerance can be used to specify the internal threshold for checking equality (any floating point number below the tolerance is treated as numerically zero).

 i1 : R = RR[x1, x2] o1 = R o1 : PolynomialRing i2 : f = 15*x1^2 + 20*x1*x2 - 36*x2^2 + 20*x1 + 16*x2 + 1 2 2 o2 = 15x1 + 20x1*x2 - 36x2 + 20x1 + 16x2 + 1 o2 : R i3 : bivariateDiagEntries f o3 = (| 19.2195 |, | 18 |, | 17 |, | 15.5926 |) | .780456 | | -2 | | 3 | | .407434 | o3 : Sequence