Polynomials with large partial sums
β Scribed by D.J Newman
- Publisher
- Elsevier Science
- Year
- 1978
- Tongue
- English
- Weight
- 153 KB
- Volume
- 23
- Category
- Article
- ISSN
- 0021-9045
No coin nor oath required. For personal study only.
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## Abstract Let __t__(__n, k__) denote the TurΓ‘n numberβthe maximum number of edges in a graph on __n__ vertices that does not contain a complete graph __K__~k+1~. It is shown that if __G__ is a graph on __n__ vertices with __n__ β₯ __k__^2^(__k__ β 1)/4 and __m__ < __t__(__n, k__) edges, then __G__
We obtain estimates of complete rational exponentials sums with sparse polynomials and rational functions f (x)=a 1 x r1 + } } } +a t x rt depending on the number of non zero coefficients t rather than on the degree.
For m Β₯ Z + let F(m) be the set of numbers with an infinite continued fraction expansion where all partial quotients, except possibly the first, do not exceed m. In 1975, James Hlavka conjectured that F( )+F( ) ] R. We shall disprove Hlavka's conjecture, showing that in fact F(5) Β± F(2)=R.