Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-17

September 2000

Equimodular Curves

Norman Biggs


This paper is motivated by a problem that arises in the study of partition functions of antiferromagnetic Potts models, including as a special case the chromatic polynomial. It relies on a theorem of Beraha, Kahane and Weiss, which asserts that the zeros of certain sequences of polynomials approach the curves on which a matrix has two eigenvalues with equal modulus. It is shown that (in general) the equimodular curves comprise a number of segments, the end-points of which are the roots of a polynomial equation, representing the vanishing of a discriminant. The segments are in bijective correspondence with the double roots of another polynomial equation, which is significantly simpler than the discriminant equation. Singularities of the segments can occur, corresponding to the vanishing of a Jacobian. These results are illustrated by explicit calculations in a specific case.

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