On A.V. Malyshev's approach to Minkowski's conjecture concerning the critical determinant of the region ∣x∣p+∣y∣p1|x|^p + |y|^p 1∣x∣p+∣y∣p1 (original) (raw)

On Minkowski's inequality and its application

Journal of Inequalities and Applications, 2011

In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.

Critical lattices, elliptic curves and their possible dynamics

We present a combinatorial geometry and dynamical systems framework for the investigation and proof of the Minkowski conjecture about critical determinant of the region $ |x|^p + |y|^p < 1, p > 1. $ The application of the framework may drastically reduce the investigation of sufficiently smooth real functions of many variables. Incidentally, we establish connections between critical lattices, dynamical systems and elliptic curves.

COUNTEREXAMPLES TO MINKOWSKI'S CONJECTURE AND ESCAPE OF MASS IN POSITIVE CHARACTERISTIC

2023

We show that there are infinitely many counterexamples to Minkowski's conjecture in positive characteristic regarding uniqueness of the upper bound of the multiplicative covering radius, µ, by constructing a sequence of compact A orbits where µ obtains its conjectured upper bound. In addition, we show that these orbits, as well as a slightly larger sequence of orbits, must exhibit complete escape of mass.

A note on the Jacobian Conjecture

Linear Algebra and its Applications, 2011

In this note, we show that, if the Druzkowski mappings F (X) = X + (AX) * 3 , i.e. F (X) = (x 1 + (a 11 x 1 + • • • + a 1n x n) 3 , • • • , x n + (a n1 x 1 + • • • + a nn x n) 3), satisfies T rJ((AX) * 3) = 0, then rank(A) ≤ 1 2 (n + δ) where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ≤ 9 in the case n i=1 a ii = 0.

On the Jacobian Conjecture

Taiwanese Journal of Mathematics

Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given by coordinate functions fl, f,, where each fi is a polynomial in n variables X1, X,. Iffhas a polynomial inverse g (g 1, ,), then the determinant of the Jacobian matrix f/OXj is a non-zero constant. This follows from the chain rule: Since f 0 is the identity, we have X 0i, f,), so X t=l This says that the product tXs is the identity matrix. Thus, the Jacobian determinant off is a non-vanishing polynomial, hence a constant. The Jacobian conjecture states, conversely, that if the characteristic of k is zero, and if f= (f,..., f,) is a polynomial map such that the Jacobian determinant is a non-zero constant, then f has a polynomial inverse. The problem first appeared in the literature (to my knowledge) in 1939 in [11] for k C. Many erroneous proofs have emerged, several of which have been published, all for k C, n 2. The conjecture is trivially true for n 1. For n > 1, the question is open. There has been a vigorous attempt by S. Abhyankar and T.-T. Moh to solve the problem for n 2. In this case it is known that the Jacobian conjecture is equivalent to the assertion that whenever f (f, f2) satisfies the Jacobian hypothesis, the total degree off divides that off2, or vice versa. Abhyankar and Moh have obtained a number of partial results by looking at the intersection of the curves fl and f2 at infinite in p2. Moh has proved, in fact, that the conjecture is true provided the degrees of A and A do not exceed 1 [15].

On The Gauss EYPHKA Theorem And Some Allied Inequalities

I use the 1907 Hurwitz formula along with the Jacobi triple product identity to understand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x 2 + 16y 2 + 36z 2 + 16yz + 4xz + 8xy and 9x 2 + 17y 2 + 32z 2 − 8yz + 8xz + 6xy. I also discuss three nontrivial analogues of the Gauss EΥPHKA theorem. My technique can be applied to all known spinor regular ternary quadratic forms.

Counterexamples to Minkowski's Uniqueness Conjecture and Escape of Mass in Positive Characteristic

arXiv (Cornell University), 2023

Some notes on de Oliveira's determinantal conjecture

Linear Algebra and its Applications, 1989

Let A, B, U EC"'", A = diag( a j), B = diag( bj), U unitary, D = det( A + UBU") , 'o= fi (aj+bo(j,), 0 E S,, j=l

Extension of Fermat’s last theorem in Minkowski natural spaces

Journal of Mathematical Chemistry

Minkowski natural (N + 1)-dimensional spaces constitute the framework where the extension of Fermat’s last theorem is discussed. Based on empirical experience obtained via computational results, some hints about the extension of Fermat’s theorem from (2 + 1)-dimensional Minkowski spaces to (N + 1)-dimensional ones. Previous experience permits to conjecture that the theorem can be extended in (3 + 1) spaces, new results allow to do the same in (4 + 1) spaces, with an anomaly present here but difficult to find in higher dimensions. In (N + 1) dimensions with N > 4$$ N > 4 there appears an increased difficulty to find Fermat vectors, there is discussed a possible source of such an obstacle, separately of the combinatorial explosion associated to the generation of natural vectors of high dimension.

On packing of Minkowski balls. II

arXiv (Cornell University), 2023

This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices are calculated.

Volumes of subset Minkowski sums and the Lyusternik region

2021

We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of M compact sets in R d , which we call the Lyusternik region, and make some first steps towards describing it. Our main result is that a fractional generalization of the Brunn-Minkowski-Lyusternik inequality conjectured by Bobkov et al. (2011) holds in dimension 1. Even though Fradelizi et al. (2016) showed that it fails in general dimension, we show that a variant does hold in any dimension. Content

Remark on generalization of Minkowski's inequality

Acta Universitatis Carolinae. Mathematica et Physica, 1995

Let (Q, X, «) be a measure space such that w(Q) < 1. We give some general conditions for a bijection <p: [0, oo) i-> [0, oo), such that for all w-integrable simple functions x, y: Q i-> R. This generalizes result from .

On the Ilmonen-Haukkanen-Merikoski Conjecture

2015

Let K_n be the set of all n× n lower triangular (0,1)-matrices with each diagonal element equal to 1, L_n = { YY^T: Y∈ K_n} and let c_n = _Z∈ L_nμ_n^(1)(Z):μ_n^(1) (Z) is the smallest eigenvalue of Z . The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that c_n is equal to the smallest eigenvalue of Y_0Y_0^T, where (Y_0)_ij= < a r r a y >. In this paper we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.

A Generalization of a Theorem of Boyd and Lawton

Canadian Mathematical Bulletin, 2012

This thesis applies to study first, in part 1, the Mahler measure of polynomials in one variable. It starts by giving some definitions and results that are important for calculating this height. It also addresses the topic of Lehmer's question, an interesting conjecture in the field, and it gives some examples and results aimed at resolving the issue. The extension of the Mahler measure to several variable polynomials is then considered including the subject of limit points with some examples. In the second part, we first give definitions of a higher order for the Mahler measure, and generalize from single variable polynomials to multivariable polynomials. Lehmer's question has a counterpart in the area of the higher Mahler measure, but with totally different answers. At the end, we reach our goal, where we will demonstrate the generalization of a theorem of Boyd-Lawton. This theorem shows a relation between the limit of Mahler measure of multivariable polynomials with Mahler measure of polynomials in one variable. This result has implications in terms of Lehmer's conjecture and serves to clarify the relationship between the Mahler measure of one variable polynomials, and the Mahler measure of multivariable polynomials, which are very different.