La mesure de Mahler supérieure et la question de Lehmer

Transcription

1 La mesure de Mahler supérieure et la question de Lehmer Matilde Lalín (joint with Kaneenika Sinha (IISER, Kolkata)) Université de Montréal mlalin Conférence Québec-Maine Université Laval 2 octobre, 2010 (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 1 / 23

2 Mahler measure for one-variable polynomials Pierce (1918): P Z[x] monic, P(x) = i (x r i ) n = i (r n i 1) P(x) = x 2 n = 2 n 1 (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 2 / 23

3 Lehmer (1933): { r n+1 1 r if r > 1 lim = n r n 1 1 if r < 1 For M(P) = a P(x) = a (x r i ) i r i, m (P) = log a + log r i. By Jensen's formula, r i >1 r i >1 m(p) := 1 0 ( log P 2πiθ) e dθ. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 3 / 23

4 Kronecker's Lemma P Z[x], P 0, m(p) = 0 P(x) = x k Φ ni (x) where Φ ni are cyclotomic polynomials. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 4 / 23

5 Lehmer's question Lehmer (1933): Given ɛ > 0, can we nd a polynomial P(x) Z[x] such that 0 < m(p) < ɛ? m(x 10 + x 9 x 7 x 6 x 5 x 4 x 3 + x + 1) = Is the above polynomial the best possible?. 379 = 1, 794, 327, 140, 357 (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 5 / 23

6 Lehmer's question Lehmer (1933): Given ɛ > 0, can we nd a polynomial P(x) Z[x] such that 0 < m(p) < ɛ? m(x 10 + x 9 x 7 x 6 x 5 x 4 x 3 + x + 1) = Is the above polynomial the best possible?. 379 = 1, 794, 327, 140, 357 (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 5 / 23

7 P C[x] reciprocal i P(x) = ±x deg P P ( x 1). Theorem (Smyth, 1971) P Z[x] nonreciprocal, m(p) m(x 3 x 1) = (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 6 / 23

8 Higher Mahler measure For k Z 0, the k-higher Mahler measure of P is m k (P) := 1 0 log k P (e 2πiθ) dθ. k = 1 : m 1 (P) = m(p), and m 0 (P) = 1. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 7 / 23

9 The simplest examples (Kurokawa, L., Ochiai) m 2 (x 1) = ζ(2) 2 = π2 12. m 3 (x 1) = 3ζ(3). 2 m 4 (x 1) = 3ζ(2)2 + 21ζ(4) 4 = 19π m 5 (x 1) = 15ζ(2)ζ(3) + 45ζ(5). 2 m 6 (x 1) = 45 2 ζ(3) π6. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 8 / 23

10 k=0 m k (x 1) s k = k! = 1 0 2πiθ e 1 s dθ ( ) Γ(s + 1) Γ ( s + 1) = exp ( 1) k (1 2 1 k )ζ(k) s k. 2 k 2 k=2 Transform the integral into a Beta function, then use the Weierstrass product of the Γ-function. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 9 / 23

11 Lehmer's question for even higher measure Theorem If P(x) Z[x], then for any h 1, ( ) π 2 h 12, if P(x) is reciprocal, m 2h(P) ( ) π 2 h 48, if P(x) is non-reciprocal. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 10 / 23

12 m 2 (P) for P reciprocal is minimized when P is a product of monomials and cyclotomic polynomials. m 2 (P) π2 for P a product of monomials and cyclotomic 12 polynomials. For P nonreciprocal, take P(x)x deg P P ( x 1 ). For m 2h(P), use Hölder's inequality. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 11 / 23

13 Multiple Mahler measure Let P 1,..., P k C[x ± ] be non-zero Laurent polynomials. Their multiple higher Mahler measure is dened by m(p 1,..., P k ) = m 2 ( i 1 0 (x r i ) ( log P 2πiθ) Pk 1 e log (e 2πiθ) dθ. ) = i,j m(x r i, x r j ). (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 12 / 23

14 Multiple Mahler measure for simple polynomials Lemma (Kurokawa, L., Ochiai) For 0 α 1 m(x 1, x e 2πiα ) = π2 2 ( α 2 α + 1 ). 6 (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 13 / 23

15 m 2 of cyclotomic polynomials Proposition For any two positive integers a and b, m(x a 1, x b 1) = π2 12 (a, b) 2 ab. For a positive integer n, let φ n (x) denote the n-th cyclotomic polynomial and ϕ Euler's function. Then m 2 (φ n (x)) = π2 12 ϕ(n)2 r(n), n where r(n) denotes the number of distinct prime divisors of n. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 14 / 23

16 m 2 of noncyclotomic polynomials P(x) m(p) m2(p) x 8 + x 5 x 4 + x x 10 + x 9 x 7 x 6 x 5 x 4 x 3 + x x 10 x 6 + x 5 x x 10 + x 7 + x 5 + x x 10 x 8 + x 5 x x 10 + x 8 + x 7 + x 5 + x 3 + x x 10 + x 9 x 5 + x x 12 + x 11 + x 10 x 8 x 7 x 6 x 5 x 4 + x 2 + x x 12 + x 11 + x 10 + x 9 x 6 + x 3 + x 2 + x x 12 + x 11 x 7 x 6 x 5 + x x 12 + x 10 + x 7 x 6 + x 5 + x x 12 + x 10 + x 9 + x 8 + 2x 7 + x 6 + 2x 5 + x 4 + x 3 + x x 14 + x 11 x 10 x 7 x 4 + x x 14 x 12 + x 7 x x 14 x 12 + x 11 x 9 + x 7 x 5 + x 3 x x 14 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x x 14 + x 13 x 8 x 7 x 6 + x x 14 + x 13 + x 12 x 9 x 8 x 7 x 6 x 5 + x 2 + x x 14 + x 13 x 11 x 7 x 3 + x Known noncyclotomic P Z[x], deg P 14, m(p) < (Mossingho: mjm/lehmer/search/) (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 15 / 23

17 m 3 Lemma (Kurokawa, L., Ochiai) m(x 1, x e 2πiα, x e 2πiβ ) = k,l 1 k,m 1 l,m cos 2π((k + l)β lα) kl(k + l) cos 2π((k + m)α mβ) km(k + m) cos 2π(lα + mβ). lm(l + m) (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 16 / 23

18 m 3 ( x n 1 x 1 ) = 32 ζ(3) ( 2 + 3n n 3 n 2 ) + 3π 2 j=1 n j ( ) cot π j n. j 2 Examples m 3 ( x 2 m 3 ( x 3 + x + 1 ) = 10 3π 3 ζ(3) + 2 L(2, χ 3). + x 2 + x + 1 ) = ζ(3) + 3π 2 L(2, χ 4). (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 17 / 23

19 Lehmer's question for odd higher measure Theorem Let P n (x) = x n 1. For h 1 xed, x 1 lim m 2h+1(P n ) = 0. n Moreover, the sequence m 2h+1(P n ) is nonconstant. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 18 / 23

20 Theorem (essentially Boyd, Lawton) Let P(x 1,..., x n ) C[x 1,..., x n ] and r = (r 1,..., r n ), r i Z >0. Dene P r (x) as P r (x) = P(x r 1,..., x rn ), and let q(r) = min H(s) : s = (s 1,..., s n ) Z n, s (0,..., 0), n s j r j = 0, j=1 Then lim m(p 1r,..., P l r ) = m(p 1,..., P l ). q(r) (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 19 / 23

21 m 2h+1 lim n m 2h+1 ( x n ) 1 x 1 (suggested by Soundararajan) ( x n ) ( ) 1 y 1 = m 2h+1 = 0. x 1 x 1 = C(h) log2h 1 n n ( ( )) 1 + O log 1 n. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 20 / 23

22 Summing up Lehmer's question FALSE for m 2h, h 1. Lehmer's question TRUE for m 2h+1, h 1. m k (P) is interesting for P cyclotomic and that answers Lehmer's question. (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 21 / 23

23 Further questions m 3 (P) for P noncyclotomic? Lehmer's question for P noncyclotomic. Bounds for m 2h(P), h 2. Explain zeta values!!! (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 22 / 23

24 Thank you! Merci! (j/w Sinha) Lalín (U de M) La mesure de Mahler supérieure Conférence Québec-Maine 23 / 23