On A.V. Malyshev's approach to Minkowski's conjecture concerning the critical determinant of the region $|x|^p + |y|^p<1$ for $p>1$

We present A.V. Malyshev`s approach to Minkowski`s conjecture (in Davis`s amendment) concerning the critical determinant of the region $|x|^p + |y|^p<1$ for $p>1$ and Malyshev`s method. In the sequel of this article we use these approach and method to present the main result.

H. Minkowski in his monograph [1] raise the question about minimum constant c such that the inequality has integer solution other than origin. Minkowski with the help of his theorem on convex body has found a sufficient condition for the solvability of Diophantine inequalities in integers not both zero: .
But this result is not optimal, and Minkowski also raised the issue of not improving constant c. For this purpose Minkowski has proposed to use the critical determinant. Given any set R ⊂ R n , a lattice Λ is admissible for R (or is R-admissible) if R Λ = ∅ or {0}. The infimum ∆(R) of the determinants (the determinant of a lattice Λ is written d(Λ)) of all lattices admissible for R is called the critical determinant of R. A lattice Λ is critical for R if d(Λ) = ∆(R).

Minkowski's conjecture as a problem of Diophantine approximation theory
Diophantine approximations connect with critical determinants and with solutions in integer numbers x 1 , . . . x n (with some restrictions, for instance not all x 1 , . . . x n are equal to zero) of inequalities [5]. Let R be a set and Λ be a lattice with base Usually in the geometry of numbers the function F (x) is a distance func- The problem of solving of diophantine inequality F (x) < c, with a distance function F has the next framework.
Let M be the closure of a set M and #P be the number of elements of a finite set P . An open set S ⊂ R n is a star body if S includes the origin of R n and for any ray r beginning in the origin #(r ∩ (M \ M )) ≤ 1. If F (x) is a distance function then the set One of the main particular case of a distance function is the case of convex symmetrical function F (x) which with conditions (i) -(iii) satisfies the additional conditions The Minkowski's problem can be reformulated as a conjecture concerning the critical determinant of the region | x | p + | y | p ≤ 1, p > 1. Recall once more that mentioned mathematical problems are closely connected with Diophantine Approximation.
For the given 2-dimension region D p ⊂ R 2 = (x, y), p > 1 : Let a ∈ Λ, a = 0 and let The Hermite constant of the function F is defined as

Moduli Spaces
What is moduli? Classically Riemann claimed that 6g − 6 (real) parameters could be for Riemann surface of genus g > 1 which would determine its conformal structure (for elliptic curves, when g = 1, it is needs one parameter). From algebraic point of view we have the following problem: given some kind of variety, classify the set of all varieties having something in common with the given one (same numerical invariants of some kind, belonging to a common algebraic family). For instance, for an elliptic curve the invariant is the modular invariant of the elliptic curve. Let B be a class of objects. Let S be a scheme. A family of objects parametrized by the S is the set of objects X s : s ∈ S, X s ∈ B equipped with an additional structure compatible with the structure of the base S. Algebraic moduli spaces are defined in the papers by Mumford, Harris and Morrison [14,15]. A possibility of the parameterization of all admissible lattices of regions D p = {|x| p + |y| p < 1}, under varying p > 1, by some analytical manifolds was mentioned in the book by Minkowski in 1907 [1]. In 1950 H. Cohn published the paper on the Minkowski's conjecture [6]. The parameterization and the corresponding analytic moduli space were one of the main tools of his approach to the investigation of the conjecture.
where σ is some real parameter; here τ = τ (p, σ) is the function uniquely determined by the conditions Definition 1. In the notation above, the surface in R 3 with coordinates (σ, p, ∆) we will called the Minkowski-Cohn moduli space.

Minkowski's analytic conjecture
In considering the question of the minimum value taken by the expression |x| p + |y| p , with p ≥ 1, at points, other that the origin, of a lattice Λ of determinant d(Λ), Minkowski [1] shows that the problem of determining the maximum value of the minimum for different lattices may be reduced to that of finding the minimum possible area of a parallelogram with one vertex at the origin and the three remaining vertices on the curve |x| p + |y| p = 1. The problem with p = 1, 2 and ∞ is trivial: in these cases the minimum areas are 1/2, √ 3/2 and 1 respectively. Let D p ⊂ R 2 = (x, y), p > 1 be the 2-dimension region: Let ∆(D p ) be the critical determinant of the region. Recall considerations of the previous section. For p > 1, let Minkowski [1] raised a question about critical determinants and critical lattices of regions D p for varying p > 1. Let Λ p , (under these conditions the lattices are uniquely defined). Using analytic parameterization Cohn [6] gives analytic formulation of Minkowski's conjecture. Let of the {p, σ} plane, where σ is some real parameter; here τ = τ (p, σ) is the function uniquely determined by the conditions In this case needs to extend the notion of parameter variety to parameter manifold. The function ∆(p, σ) in region M determines the parameter manifold.

Nikolaj Glazunoṽ
For investigation of properties of function ∆(p, σ) which are need for proof of Minkowski's conjecture [1,6] we considered the value of ∆ = ∆(p, σ) and its derivatives ∆
A compact closed interval I = [a, b] is the set of real numbers x such that (s.t.) a ≤ x ≤ b. Interval analysis with this type of intervals uses usually two sorts of intervals. Wide intervals are used for representing uncertainty of the real world or lack of information. Narrow intervals are used for rounding error bounds. In any of these two cases on each step of an interval computation we compute the interval I which contains an (ideal) solution of our problem. Some examples of implementations of the intervals are given in papers [22, ?].
There are many numerical algorithms for solving mathematical problems. The majority of these algorithms are iterative, so, since stopping the algorithms after a certain number of steps, we only get an approximationx to the desired solution x. A perfect solution would if we could estimate the errors of the result not after the iteration process, but simultaneously with the iteration process. This is one of the main ideas of interval analysis [16,18,17,19].

First present each of the expressions ∆
Then by the implicit function theorem computing τ = τ (p, σ) by means of the following iteration process: For computation of the expression for τ p we apply the following iteration: So we really have represented the function ∆ as the function ∆(p, σ) of two variables. The same fact is true for it's derivatives.
However, the spread of the approach on the domain p < 6 met with great difficulties. To prove this hypothesis in the domain 1 < p < 6 Malyshev proposed, firstly, to extend the class of functions.
It needs to add derivatives of the functions l(p, σ), g(p, σ), h(p, σ) which in themselves are quite complicated.
These are derivatives with respect to σ and with respect to p. Secondly, Malyshev proposed to construct the interval evaluation of the functions on small intervals, covering the study area.
6.1. Interval evaluation of functions. Let X = (x 1 , · · · , x n ) = ([x 1 , x 1 ], · · · , [x n , x n ] be the n-dimensional real interval vector with x i ≤ x i ≤ x i ("rectangle" or "box"). The interval evaluation of a function G(x 1 , · · · , x n ) on an interval X is the interval [G, G] such that for any x ∈ X, G(x) ∈ [G, G]. The interval evaluation is called optimal if G = min G, and G = max G on the interval X. In the case of the formula that expressing ∆ in terms of a sum of derivatives of "atoms" . .) one applies the rational interval evaluation to construct formulas for lower bounds and upper bounds of the functions, which in the end can be expressed in terms of p, p, σ, σ, τ , τ , ; here the bounds τ , τ , are obtained with the help of the iteration process: i = 0, 1, · · · As interval computation is the enclosure method, we have to put: N is computed on the last step of the iteration.
6.2. Algorithms and software modules. Here we give names, input and output of algorithms and and software modules for interval evaluation only. All these algorithms and and software modules are implemented, tested and applied under the computer-assisted proof of Minkowski's conjecture [8,9,11,12,13] .
First two algorithms are auxiliary and described in [22].

Algorithm MonotoneFunction
Input: A real function F (x, y) monotonous by x and by y.

Conclusions
A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning the critical determinant of the region |x| p + |y| p < 1 for p > 1 is proposed and A.V. Malyshev's method of its prove is given. Applications of the approach and of the method are presented.