Perron number

In mathematics, a Perron number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements, other than its complex conjugate, are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial $x^{2}-3x+1$ is a Perron number.

Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic coefficients whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix.

Any Pisot number or Salem number is a Perron number, as is the Mahler measure of a monic integer polynomial.

References

• Borwein, Peter (2007). Computational Excursions in Analysis and Number Theory. Springer Verlag. p. 24. ISBN 0-387-95444-9.