Computing the measure of a polynomial. (English) Zbl 0629.12002

Let \(P(x)\) be a polynomial of degree \(d\) with integer coefficients, leading coefficient \(a_0\) and roots \(x_1,\dots,x_d\). This paper considers some methods of computing \(\#(P)\), the number of roots of \(P\) satisfying \(| x_i| >1\), and \(M(P)\), the Mahler measure of \(P\), defined by
\[ M(P) = | a_0| \prod \max (| x_i|,1) = \exp \biggl(\int_0^1\log | P(e^{2\pi it})| \,dt \biggr). \]
The root-squaring algorithm for computing \(M(P)\) which is discussed in section 2.2 was introduced in a paper of the reviewer [Math. Comput. 35, 1361–1377 (1980; Zbl 0447.12002)], where a discussion of its advantages and disadvantages is given (see esp. p. 1367). Two other natural methods are suggested by the two formulas for \(M(P)\) given above. For the authors’ example on p. 31, a short computation produces the root of 12 decimal place accuracy giving \(M(P)=7.0436280134\) and \(\#(P)=2\). The results of V. Pan [Comput. Math. Appl. 14, 591–622 (1987)], for example, allow a complexity analysis of this approach. It would be interesting to analyse an approach based on numerical integration of \(\log | P(e^{2\pi it})|\), which has been used successfully by C. J. Smyth.
The authors indicate an algebraic method based on exterior products for computing \(P_k(x)\), the polynomial whose roots are the products of the form \(x_{i_1}\dots x_{i_k}\), \(1\leq i_1< \dots <i_k\leq d\). Knowing \(\#(P)\), this gives a polynomial of which \(M(P)\) is a root. They state this “this result is mostly of theoretical interest”. A more practical approach is to use the power sums of \(P\) and \(P_k\) and Newton’s formulas.


11R09 Polynomials (irreducibility, etc.)
11Y40 Algebraic number theory computations


Zbl 0447.12002
Full Text: DOI


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