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A sharp estimate of the number of integral points in a 4-dimensional tetrahedra. (English) Zbl 0844.11063
There 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 $$a= b= c= d= \text{integer}$$.

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)
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