The distribution of values of Mahler's measure (Q2754267)

Let \({\mathcal M}(d,T)\) be the number of integer polynomials of degree at most \(d,\) and of Mahler's measure at most \(T.\) The problem of estimating \({\mathcal M}(d,T)\) when \(d\) is large compared with \(T\) is quite difficult. Some estimates from above were obtained by the reviewer and \textit{S. V. Konyagin} [Acta Arith. 86, No. 4, 325-342 (1998; Zbl 0926.11080)], but they are apparently far from being sharp when, for instance, \(T\) is fixed. This paper deals with the case when \(T\) is large compared with \(d.\) Then \({\mathcal M}(d,T)\) is approximately equal to the volume of the set of polynomials with \textit{real} coefficients of degree \(\leq d\) and of Mahler's measure \(\leq T,\) which leads to the problem of estimating the number of integer lattice points in certain nonconvex, symmetric star body. The authors not only determine the volume of this star body (which turns out to be a rational number), but also find asymptotic and explicit estimates for the remainder term. NEWLINENEWLINENEWLINEMore precisely, let \(D=[(d-1)/2]\) throughout, where \([\dots]\) is the integral part of a number, and NEWLINE\[NEWLINEV_{d+1}=2^{d+1}(d+1)^D \prod _{k=1} ^{D} (2k)^{d-2k} (2k+1)^{2k-1-d}.NEWLINE\]NEWLINE It is shown that, for each fixed positive integer \(d,\) we have \({\mathcal M}(d,T)=V_{d+1} T^{d+1} + O_d(T^d)\) as \(T \to \infty.\) Furthermore, if \(d \geq 2\) and \(T \geq 8^d d^{2d},\) then NEWLINE\[NEWLINE|{\mathcal M}(d,T)-V_{d+1}T^{d+1}|\leq 8d^2 V_{d+1} T^{d+1-1/d}.NEWLINE\]NEWLINE Similar results for the number of integer monic polynomials of degree equal to \(d,\) and of Mahler's measure at most \(T,\) are also obtained. NEWLINENEWLINENEWLINEThese remarkable results were derived by finding the Lebesgue measure of the set of points \((a_{d-1},\dots,a_1,a_0) \in {\mathbb{R}}^d\) for which the Mahler measure of the polynomial \(x^d+a_{d-1}x^{d-1}+ \dots + a_1 x + a_0\) is less than or equal to \(\xi,\) where \(1 \leq \xi < \infty,\) which is shown to be equal to NEWLINE\[NEWLINE(D !)^{-1} \prod _{k=1} ^{D} (1+1/2k)^{2k-d} \sum _{j=0} ^{D} (-1)^j (d-2j)^D {D \choose j} \xi^{d-2j}.NEWLINE\]NEWLINE In particular, this implies that the volume of the set of polynomials with \textit{real} coefficients of degree \(\leq d\) and of Mahler's measure \(\leq 1\) is equal to \(V_{d+1}.\) (A similar formula for the volume of the set of real polynomials with all roots on the unit circle was earlier obtained by \textit{S. A. DiPippo} and \textit{E. W. Howe} [J. Number Theory 73, No. 2, 426-450 (1998; Zbl 0931.11023)].) NEWLINENEWLINENEWLINEIn passing, the authors prove the following result, which is of independent interest: if \(P\) and \(Q\) are two polynomials with complex coefficients, both of degree at most \(d,\) then NEWLINE\[NEWLINE|M(P)^{1/d}-M(Q)^{1/d}|\leq 2 L(P-Q)^{1/d},NEWLINE\]NEWLINE where \(M\) stands for the Mahler measure and \(L\) stands for the length of a polynomial. It would be of interest to find a similar inequality for polynomials in several variables.