By K. Alladi, M. L. Robinson (auth.), Melvyn B. Nathanson (eds.)
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Additional resources for Number Theory Carbondale 1979: Proceedings of the Southern Illinois Number Theory Conference Carbondale, March 30 and 31, 1979
Sample text
This notion was introduced into transcendence theory by Chudnovsky to extend Gelfond's method to show the algebraic independence of n+l numbers in the sets SI, $2, S 3. ~'=0 bj x3 For two polynomials f(x) =~m=0 aixl = = am~ii=l(X - s i) and g(x) = n bn~j=l(X - tj ), their semi-resultant is defined to be r(f,g) = amnbnm Z(s i - tj), where the product is taken over all pairs (i,j) with s. # t.. z j that r(f,g) # 0, one can show [Ch desirable properties of resultants. While it is clear 2], [Br 6] that semi-resultants have many of the $4 LEMMA.
COLORED SEQUENCES. > 2. 2. M + N + ~N, then tr. deg. K K (S 3) > 2. Chudnovsky introduced this somewhat fanciful name to describe part of the procedure involved in establishing an alternate generalization of Gelfond's criterion [Ch 4 ]. Waldschmidt gave a development part on notes from a lecture of mine at Cambridge University. further refinement of Chudnovsk~'s criterion. ~i ~ T2 + I. Generally [Wald 4] based in Here we present a For simnlicity we assume that the exoression ~i + 3T2 - I in the statement must be replaced by max {4T2, T I + 3T 2 - I}.
Number Th. [Br 4] 6 (1974), 22-31. , Gelfond's method for algebraic independence, Trans. S. 210 (1975), 1-26. [Br 5] , Pairs of polynomials small at a number to certain i algebraic powers, Sem. Delange Pisot Poitou, 17e annee (1975/76), No. ii [Br 6] , Some remarks on semi-resultants, Chapter 14 in Transcendence Theory: Advances and Applications, A. W. Masser, eds, Academic Press, London, 1977. [Br 7] , On the Gelfond-Feldman measure of algebraic independence, Compositio Math. D. Brownawell and M.