VG1 .- Conjecture of Lehmer, Conjecture of Schinzel-Zassenhaus 1 / 49 Obj. : to study the problem of the minoration of the Mahler measure M(a) of a nonzero algebraic number a, which is not a root of unity, in higher dimension the minoration of the height h(P) of a rational point P of an algebraic variety, by means of the dynamical zeta function zb (z) of the b-shift. 2 / 49 1.– The Mahler measure of a polynomial To study solutions of Diophantine equations it is natural to try to associate to such solutions a “size”, a ”complexity” or a ”measure”. Any computation on partial sets of solutions then takes advantage of some finiteness properties when this size lies below some bound. The first step consists in attributing a real value to an algebraic number, to solve the problem of finding such a “size” for algebraic numbers. An algebraic number a is described uniquely by its minimal polynomial Pa (X), or say f 2 Z[X]. It has integer coefficients, which are relatively prime and by convention its dominant coefficient is positive. It is natural to establish several notions of “size” on polynomials. 3 / 49 Let f 2 C[X] be a nonzero polynomial with complex coefficients of degree d : d d d−1 f = a0X + a1X + :::ad = a0 ∏(X − ai ): i=1 Several notions of height for f ; at least : (i) the Mahler measure of f : d M(f ) = ja0j ∏ maxf1;jai jg; i=1 (ii) the usual height of f : H(f ) = maxfja0j;ja1j;:::;jad jg; (iii) the Euclidean norm of f : Z 1 1=2 2 2 2 1=2 2ipt 2 L2(f ) = (ja0j + ja1j + ::: + jad j ) = jf (e )j dt ; 0 4 / 49 (iv) the sup norm on the unit disc (or on the unit circle, which is the same by the maximum modulus principle) : jf j1 = sup jf (z)j = sup jf (z)j; jzj≤1 jzj=1 (v) the length of f L(f ) = ja0j + ja1j + ::: + jad j; (vi) Bombieri’s norm of f (for factorization algorithms) : !1=2 d ja j2 [f ] = i ∑ d i=0 i Inequalities : −1=2 d (d + 1) M(f ) ≤ H(f ) ≤ L2(f ) ≤ jf j1 ≤ L(f ) ≤ 2 M(f ): 5 / 49 and, from the symmetric functions of the roots d ja j ≤ M(f )(Norm inequality): i i [recall a = (−1)d−i a ::: ] i 0 ∑ ai1 aid−i i1;:::;id−i ⊂f1;:::;dg If f is monic and has roots only in D(0;1), then M(f ) ≤ [f ] since [f ] ≥ 1. The same holds if all the roots of f are outside D(0;1). So if we want [f ] ≤ M(f ) some roots should be inside some outside D(0;1). 6 / 49 Theorem (Degot´ - Jenvrin, ’98) Let f 2 C[X] be monic, deg(f ) = d decomposed as d = n1 + n2 + n3, where n1 = number of roots inside D(0;a), 0 < a < 1, n2 = number of roots outside D(0;b), b > 1, and where n1;n2;n3;d;a;b satisfy : 2 n1 1 n1 1 + a n2 1 + ab n3 1 + a (1 + ab)n1 (1 + )n2 (1 + b)n3 × + + b 2 d 1 + ab d 1 + b 2 d 1 + b n 1 + a n 1 + b n 2 n3 × 1 + 2 + 3 ≤ 1; d 1 + ab d 1 + b 2 d 1 + b then [f ] ≤ M(f ): 7 / 49 Theorem (Jensen’s formula) For any f 2 C[X], f 6≡ 0, Z 2p M(f ) = exp Logjf (eit )jdt 0 Pf. The logarithm is additive. It suffices to consider f (z) = z − a. 3 cases : jaj > 1 : the function Logjf (z)j is harmonic in a neighbourhood of the unit circle ; hence Logjf (0)j = Logjaj: jaj < 1 : g(z) = 1 − az has no zero on jzj = 1, and Logjg(z)j is harmonic in a neighbourhood of jzj = 1. We have : jf (z)j = jg(z)j on jzj = 1, hence Z 2p Z 2p Logjf (eit )jdt = Logjg(eit )jdt = Logjg(0)j = 0: 0 0 8 / 49 Pf. jaj = 1 : it comes from 1 Z 2p 1 Z 2p Logjeit − ajdt = Logjeit − 1jdt = 0: 2p 0 2p 0 Definition The logarithmic Mahler measure of an algebraic number a is LogM(a), and denoted by m(a): 9 / 49 Definition The Mahler measure of an algebraic number a is the Mahler measure of its minimal polynomial Pa (X) 2 Z[X]. If P(X) = a0(X − a1)(X − a2):::(X − an) = n n−1 a0X + a1X + ::: + an−1X + an 2 C[X]; a0an 6= 0 then M(P) := ja0j ∏ jai j and M(P) ≥ ja0j: i;jai j>1 The smallest Mahler measures to be considered when P has integer coefficients : for ja0j = 1, i.e. over algebraic integers. 10 / 49 Properties : (i) multiplicativity : P = P1 × P2 × ::: × Pm ) M(P) = M(P1) :::M(Pm). M(a1 :::am) = M(a1):::M(am); (ii) if P∗(X) := X deg(P)P(1=X) is the reciprocal polynomial of P, then M(P∗) = M(P); in particular, for any nonzero a algebraic number M(±a±1) = M(a); q d q q(d+1) 1=q (iii) if Pq(z) = a0 ∏j=1(z − aj ) = (−1) ∏z;z d =1 P(z z), then, for any norm on C[X]q, 1=q lim kPqk = M(P); q!¥ (iv) for a any root of unity M(a) = 1: 11 / 49 Theorem (Kronecker, 1857) Let a be an algebraic integer. Then M(a) = 1 () a is a root of unity, or a = 0: Pf. =) : let f be the minimal polynomial of a. Since M(fq) = 1, the (q) coefficient aj of fq satisfies (from the Norm inequality) : d ja(q)j ≤ : j j Thus the set of such fq is finite, and the set of all roots of all fq is finite. Consequently, ak = ar for some k > r. We deduce ak−r = 1 or a = 0. 12 / 49 Definition b is a Pisot number if it is an algebraic integer (real) > 1 and if all its (Galois) conjugates jb (i)j < 1. b > 1 is a Salem number if jb (i)j ≤ 1 with at least one conjugate on jzj = 1. b and 1=b are conjugated with pairs of complex conjugated roots of modulus 1. A Perron number b ≥ 1 is an algebraic integer for which either b = 1 or jb (i)j < b if b > 1. Denote PPerron := fPerrong. The set PPerron is dense in [1;¥). The notations are usual : S:= fPisotg;T:= fSalemg, Theorem (Adler Marcus, ’79 - Memoirs of the AMS Vol.20, No 219) For any algebraic number a, the Mahler measure M(a) is a Perron number. 13 / 49 “Topological entropy and equivalence of dynamical systems”, uses the theory of Perron-Frobenius. ... in which the topological entropy of an automorphim of the d-dimensional torus = LogM(P)... Inclusions : S ⊂ PPerron T ⊂ PPerron fM(a) j a alg. numberg ⊂ PPerron; fM(P) j P 2 Z[X]g ⊂ PPerron: The last two inclusions are strict (Dubickas 2004, Boyd 1981) : for example, for m > 3, the Perron number b of minimal polynomial −1 − x + xm is not a Mahler measure. 14 / 49 Theorem (Dubickas, ’04) If b is a Perron number then nb is a Mahler measure for some n 2 N. Proof : Assume that b has minimal polynomial P(x) of degree d with ∗ leading coefficient b 2 N and with na¨ıve height H(b). Let b be the maximal modulus of all roots of P different from b (if d = 1, set b ∗ = 0). Since b is a Perron number, b ∗ < b. There exists infinitely many pairs of prime numbers q 6= t for which b ∗ < q=(tb) < b. Consider such a pair (q;t) such that q > H(b). Then we claim that nb, where n = qd−1tb, is the Mahler measure of tbb=q. Indeed tbb=q is the root of the polynomial P(qx=(tb)). Furthermore, the polynomial td bd−1P(qx=(tb)) has integer coefficients. Its two extreme coefficients are qd and td bd−1P(0). These two integers are relatively prime. Thus td bd−1P(qx=(tb)) is an integer irreducible polynomial. All its roots, except for tbb=q lie in the unit circle. Then M(tbb=q) = qd tbb=q = qd−1tbb = nb: 15 / 49 Usual notations : ∗ A polynomial P 2 C[X] such that P = P is said reciprocal. The minimal polynomial of a Salem number is monic and reciprocal i.e. P∗ = P. By convention, a Salem polynomial denotes the product of cyclotomic polynomials by the minimal polynomial of a Salem number. The minimal polynomial of a Pisot number is nonreciprocal. A Pisot polynomial is the minimal polynomial of a Pisot number ; it is monic, irreducible and nonreciprocal. A Perron polynomial is the minimal polynomial of a Perron number. Open question (explicit algebraicity of the Mahler measure) : For any algebraic number a what is the minimal polynomial of M(a) ?, i.e. what is the Perron polynomial N−1 N g(X) = b0 + b1X + ::: + bN−1X + X 2 Z[X] such that g(M(a)) = 0? 16 / 49 What are the relations between N, d = deg(a), the bj s and the coefficients ai of the minimal polynomial of a ? If a is a Pisot number or a Salem number, then M(a) = a; N = d, and the coefficient vector of g is exactly the coefficient vector of the minimal polynomial. Open question (“inverse problem for Mahler measures”) : For any Perron number b 2 PPerron is b the Mahler measure M(a) of some algebraic number a, i.e.