The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke’s theorem for infinite dimensional Hopf algebras. The generalization of Maschke’s theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

Abstract When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$ , we consider the system of Diophantine equations \begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*} We show that in this cousin a Vinogradov system, there is paucity non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o\!\le...

Given a family $$\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$$ of d distinct nonconstant polynomials, positive integer $$k\le d$$ and real parameter $$\rho$$ , we consider the mean value $$M_{k, \rho} (\varphi, N) \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; |^\rho \,d{\rm $$ of exponential sums $$S_{\varphi}({\rm \sum_{n=1}^{N} \...