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.

MSC:

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

Zbl 0447.12002
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References:

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