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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

Citations:

Zbl 0447.12002
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References:

[1] Cantor, D.C.; Straus, E.D., On the conjecture of D. H. Lehmer, Acta arith., 42, 97-100, (1982) · Zbl 0504.12002
[2] Ceriienco, L.; Delogu, G.; Piras, F., Prodotti esterni di s.r.l.e metodi per la ricerca approssimata delle radici di un polinomio, Rend. sem. fac. sci. Cagliari, 50, suppl., 177-191, (1980)
[3] Cerlienco, L.; Delogu, G.; Piras, F., Successioni ricorrenti lineari del secondo ordine ed alcune loro applicazioni, Bull. math. SAC. sci. math. R.S. roumanie tome, 27, 75, 2, (1983), 111-119 · Zbl 0527.33003
[4] Cerlienco, L.; Mignotte, M.; Piras, F., Suites recurrences linéaires. propriétés algébriques et arithmétiques, L’enseignement math, (1987), (in press) · Zbl 0626.10008
[5] Dobrowolsky, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta arith., 34, 391-401, (1979) · Zbl 0416.12001
[6] Durand, E., Solutions numerulues des equations algébriques, (1960), Masson & Cie Paris
[7] Dym, H.; McKean, H.P., Fourier series and integrals, (1972), Academic Press London · Zbl 0242.42001
[8] Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1934), Cambridge University Press Cambridge
[9] Henrici, P., ()
[10] Kronecker, L., Zwei Sätze über gleichungen mit ganzzahligen coefficienten, J. für reine and angew. math., 53, 173-175, (1857)
[11] Landau, E., Sur quelques theorémes de M. petrovic relatifs aux zéros des functions analytiques, Bull. soc. math. France, 33, 251-261, (1905) · JFM 36.0467.01
[12] Lawton, W., Heights of algebraic numbers and szegö’s theorem, Proc. amer. math. soc., 49, 47-50, (1975) · Zbl 0306.12002
[13] Lehmer, D.H., Factorization of certain cyclotomic functions, Ann. math., 34, 461-479, (1933) · Zbl 0007.19904
[14] Louboutin, R., Sur la mesure de Mahler des nombres algébriques, C.R. acad. sci. Paris, 296, 707-708, (1983) · Zbl 0557.12001
[15] Marden, M., The geometry of the zeros of a polynomial in a complex variable, (1949), American Mathematics Society New York · Zbl 0038.15303
[16] Mignotte, M., An inequality about factors of polynomials, Math. comp., 28, 1153-1157, (1974) · Zbl 0299.12101
[17] Mignotte, M., Entiers aglébriques dont LES conjugués sont proches du cercle unité, Sem. delangepisot-poitou, 39, 1-6, (1977-78)
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