Cinquante Ans de Polynomes Fifty Years of Polynomials. Proc. by Michel Langevin, Michel Waldschmidt

By Michel Langevin, Michel Waldschmidt

Ahead of his premature dying in 1986, Alain Durand had undertaken a scientific and in-depth research of the mathematics views of polynomials. 4 unpublished articles of his, shaped the center-piece of recognition at a colloquium in Paris in 1988 and are reproduced during this quantity including eleven different papers on heavily comparable themes. a close advent by way of M. Langevin units the scene and locations those articles in a unified standpoint.

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C) If(k,1)EBAP then 1-k-1/2<0. (d) AP n BAP = 1(1/6,2/3)1. Proof. \) or (k, l) = BA(ic, \) for some (rc, A) E P. In either case, k+1- 2rc+\+1 2rc+2 <1. Optimal Exponent Pairs 56 This proves (a). To prove (b), suppose (k, 1) = A(a, A). 1) since A > 1/2 > ic. Part (c) follows from (a) and (b) and the observation that B is a reflection through the line 1 - k - 1/2. 1) if and only if (a, A) = (1/2,1/2). Note that (1/6,2/3) = AB(0,1) = BAB(0,1). In geometric terms, (a) states that P lies in the triangle with vertices (0, 1), (0, 1/2), and (1/2,1/2).

If n > v then A(n) _ - E A(k) E µ(d) + E µ(k) log 1- E A(k)y(m). kl=n k>v,I>u dal d v. 12 by e(jn1) and summing over n, we get E A(n)e(jn7) 1 b(m)e(jm1'n-') 1

Up_1)F(ul+1)(x,,) ... 1), we see that If'(xv) - F'(xv)I = IV - yx-'I < Eyx'. Consequently, (1 - e)°rIV ° < x < (1 + e)°,1V °, and so Ixv - XYI << ei v-°. 1), we see that if 0 < p < P - 1 then fP+'(xv) - FP+1(xv)I I < EIFP+1(xv)I << EyN-'-P Furthermore, there is some tin the open interval with endpoints x and X such that IFP+1(xi) - FP+1(Xv)I = IF(P+21(t)(X,, - xY)I << EyN-'-P-1nv ° «EyN-'-P. )I «EyN "-P. 2), and thus completes the proof of the lemma. 9 says that if f E F(N, P, s, y, e) then the restriction of 0 to the interval [a,,6] fl [J, 2J] is in F(J, P, o, q, CE) for any J, a

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