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A remark on an algorithm for testing Pisot polynomials. (English) Zbl 1463.11096

A Pisot polynomial is, by definition, a monic polynomial with integer coefficients with nonzero constant term and having exactly one root of absolute value \(>1\). These polynomials have been studied for a long time. They are interesting already for the fairly simple reason that they are all irreducible (because there is no monic polynomial with integer coefficients whose roots all have absolute value \(<1\); just look at the constant coefficient). But it is not obvious how to test the Pisot property quickly. In this short note, the author offers a certain twist on a test by R. J. Duffin [SIAM Rev. 11, 196–213 (1969; Zbl 0175.09801)]. The general drift is similar, using so-called Schur polynomials (that is, polynomials over the complex numbers whose roots are all of absolute value less than 1). There is a recursion over the degree which allows to test the Schur property rather efficiently. Both Duffin and the author begin their algorithm by an initial step that turns the given (Pisot?) polynomial into another polynomial of which one has to prove the Schur property. But the author does this differently, with the effect that a case distinction needed by Duffin is eliminated. The author concludes his note with three examples. In the last example (admittedly with a bit of luck) his method is much faster than Duffin’s. It is nice to see that these very old concepts still give rise to new ideas. The main tools are two classical theorems about holomorphic functions, due to Rouché and Hurwitz. Rouché’s theorem says: If \(f\) and \(g\) are holomorphic on an open set including the domain \(D\) and its boundary \(B\), and “\(g\) has smaller values than \(f\) on \(B\)”, then \(f\) and \(f+g\) have the same number of zeros in \(D\). The charming thing is that it is not necessary (and indeed difficult) to control exactly how the roots move when \(f\) is replaced by \(f+g\).

MSC:

11C08 Polynomials in number theory
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Citations:

Zbl 0175.09801
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