Lattice points in multidimensional bodies (Q2703551)

Let polynomials \(Q(x)= \lambda_1 x_1^p+\cdots+ \lambda_d x_d^p\), \(x\in \mathbb{R}^d\), be given with positive coefficients \(\lambda_j\) and an integer \(p\geq 2\). Consider the region \(E_s=\{x\in \mathbb{R}^d: Q(x)\leq s\}\). Let \(\text{vol}_{\mathbb{Z}} E_s\) denote the number of lattice points in \(E_s\) and \(\operatorname {vol}E_s\) the volume of \(E_s\). Then we have for the lattice rest \(\delta(s,Q)\) the estimation NEWLINE\[NEWLINE\delta(s,Q)=|\text{vol}_{\mathbb{Z}}E_s- \operatorname {vol}E_s|\ll s^{d/p-1}.NEWLINE\]NEWLINE We say that a polynomial \(Q\) is rational if there exists a real number \(M>0\) such that all \(M\lambda_j\) are natural numbers. Otherwise \(Q\) is called irrational. Then \(\delta(s,Q)= o(s^{d/p-1})\) if and only if \(Q\) is irrational. NEWLINENEWLINENEWLINEFurther, the authors introduce polynomials \(Q(x)= \sum_{i=1}^p Q_j(x)\), \(\deg Q_j=j\), \(x\in \mathbb{R}^d\), where \(Q_j\) is the \(j\)th homogeneous part of \(Q\). In particular, it is assumed that the polynomial \(Q_p(x)\) is of the form \(Q_p(x)= \sum_{j=1}^{d_0} \lambda_j x_j^p+ P_p(x)\), where \(P_p\) is a homogeneous polynomial of degree \(P_p=p\), but with degree in the variables \(x_1,\dots, x_{d_0}\) strictly smaller than \(p\). Here we have again the estimation NEWLINE\[NEWLINE\delta(s,Q)\ll s^{d/p-1},NEWLINE\]NEWLINE which is optimal provided that the dimension \(d\) is sufficiently large. NEWLINENEWLINENEWLINESome further interesting investigations and estimations cannot be described here, because they are too complicated.