By K. Alladi, M. L. Robinson (auth.), Melvyn B. Nathanson (eds.)

**Read or Download Number Theory Carbondale 1979: Proceedings of the Southern Illinois Number Theory Conference Carbondale, March 30 and 31, 1979 PDF**

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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.