A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.

*(English)*Zbl 0844.11063There are at least two reasons to consider the natural problem of estimating the number \(P_n\) of positive integral points satisfying
\[
{{x_1} \over {a_1}} + {{x_2} \over {a_2}} +\cdots+ {{x_n} \over {a_n}} \leq 1, \tag \(*\)
\]
where \(a_1\geq a_2\geq \cdots\geq a_n\geq 0\) are positive real numbers. Firstly, this is an extremely important question in number theory; it would have many applications to current problems in analytic number theory, primality testing and in factoring. Secondly the problem has an interesting application in geometry. Let \(f: ({\mathcal C}^n, 0)\to ({\mathcal C}, 0)\) be the germ of complex analytic functions with an isolated critical point at the origin. The Milnor number \(\mu\) of the singularity is \(\dim {\mathcal C} \{z_1, \dots, z_n\}/ (f_{z_1}, \dots, f_{z_n})\). Let \(\pi: M\to V\) be a resolution of \(V= \{(z_1, \dots, z_n): f(z_1, \dots, z_n) =0\}\). The geometric genus \(p_g\) of the singularity \((V, 0)\) is the dimension of \(H^{n-1} (M,{\mathcal O})\). In 1978, Durfee has made the following conjecture. Durfee conjecture. \(n! p_g\leq \mu\) with equality only when \(\mu =0\).

The connection between the Durfee conjecture and the proposed problem is as follows. A polynomial in \((z_1, \dots, z_n)\) is weighted homogeneous of type \((w_1, \dots, w_n)\), where \(w_1, \dots, w_n\) are fixed positive rational numbers if it can be expressed as a linear combination of monomials \(z_1^{i_1}, \dots, z_n^{i_n}\) for which \(i_1/ w_1+ \cdots+ i_n/ w_n=1\). If \(f\) is a weighted homogeneous polynomial of type \((a_1, \dots, a_n)\) with isolated singularity at the origin, then Milnor and Orlik proved that \(\mu= (a_1- 1) (a_2- 1) \cdots (a_n- 1)\). On the other hand, Merle and Teissier showed that \(p_g\) is exactly the number \(P_n\) of positive integral points satisfying \((*)\). Thus the Durfee conjecture provides guidance for the upper estimate of \(P_n\). The purpose of this paper is to prove the following theorem: Let \(a\geq b\geq c\geq d\geq 2\) be real numbers. Let \(P_4\) be the number of positive integral points satisfying \(x/a+ y/b+ z/c+ w/d\leq 1\). Then \[ 24P_4\leq abcd- {\textstyle {3\over 2}} (abc+ abd+ acd+ bcd)+ {\textstyle {11\over 3}} (ab+ ac+ bc)- 2(a+ b+ c), \] and the equality is attained if and only if \(a= b= c= d= \text{integer}\).

The connection between the Durfee conjecture and the proposed problem is as follows. A polynomial in \((z_1, \dots, z_n)\) is weighted homogeneous of type \((w_1, \dots, w_n)\), where \(w_1, \dots, w_n\) are fixed positive rational numbers if it can be expressed as a linear combination of monomials \(z_1^{i_1}, \dots, z_n^{i_n}\) for which \(i_1/ w_1+ \cdots+ i_n/ w_n=1\). If \(f\) is a weighted homogeneous polynomial of type \((a_1, \dots, a_n)\) with isolated singularity at the origin, then Milnor and Orlik proved that \(\mu= (a_1- 1) (a_2- 1) \cdots (a_n- 1)\). On the other hand, Merle and Teissier showed that \(p_g\) is exactly the number \(P_n\) of positive integral points satisfying \((*)\). Thus the Durfee conjecture provides guidance for the upper estimate of \(P_n\). The purpose of this paper is to prove the following theorem: Let \(a\geq b\geq c\geq d\geq 2\) be real numbers. Let \(P_4\) be the number of positive integral points satisfying \(x/a+ y/b+ z/c+ w/d\leq 1\). Then \[ 24P_4\leq abcd- {\textstyle {3\over 2}} (abc+ abd+ acd+ bcd)+ {\textstyle {11\over 3}} (ab+ ac+ bc)- 2(a+ b+ c), \] and the equality is attained if and only if \(a= b= c= d= \text{integer}\).

Reviewer: Y.-J.Xu, S.-T.Yau (Chicago)

##### MSC:

11P21 | Lattice points in specified regions |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

11A41 | Primes |

11A51 | Factorization; primality |

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |