In mathematics, Hua's lemma,[1] named for Hua Loo-keng, is an estimate for exponential sums.
It states that if P is an integral-valued polynomial of degree k,
is a positive real number, and f a real function defined by
![{\displaystyle f(\alpha )=\sum _{x=1}^{N}\exp(2\pi iP(x)\alpha ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/268074afb3a054ed42bc072c6244028a208b9085)
then
,
where
lies on a polygonal line with vertices
![{\displaystyle (2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529f209328cf78d7b474332433952c7585e9c292)
References
- ^ Hua Loo-keng (1938). "On Waring's problem". Quarterly Journal of Mathematics. 9 (1): 199–202. doi:10.1093/qmath/os-9.1.199.