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On Vinogradov’s mean value theorem. (English) Zbl 0769.11036

This paper brings to the reader’s attention the important role played by iterative methods in the context of Vinogradov’s mean value theorem. Once this has been observed, it is possible to improve substantially upon the relevant estimates by adopting the “efficient differencing” introduced by the author [Ann. Math., II. Ser. 135, 131-164 (1992; Zbl 0754.11026)]. In the present context one is interested in upper bounds for the number \(J_{k,s}(P)\) of solutions to the system of diophantine equations \[ x^ l_ 1+\cdots+ x^ l_ s=y_ 1^ l+\cdots+ y^ l_ s\qquad (1\leq l\leq k) \] with \(1\leq x_ i\), \(y_ i\leq P\). The precise bounds obtained here are rather complicated to state and we refer to the paper for details. We mention two out of various applications: The asymptotic formula in Waring’s problem for \(k\)-th powers holds if the number of variables exceeds \(\bigl(2+o(1)\bigr)k^ 2 \log k\). This beats a result of Hua Lookeng (dating back to 1949) by a factor 2. Similarly, an asymptotic formula for \(J_{k,s}(P)\) holds when \(s\geq\bigl({5\over 3}+o(1)\bigr)k^ 3 \log k\). This improves the previous best by a factor 5/9.

MSC:

11P05 Waring’s problem and variants
11D72 Diophantine equations in many variables

Citations:

Zbl 0754.11026
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References:

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