Minkowski's Theorem Research Articles

Pseudo-cones

Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the interior. The purpose of the following is to study this analogy in greater detail. We supplement the investigation of copolarity, considering, for example, conjugate faces. Then we deal with the question suggested by Minkowski's theorem, asking which measures are surface area measures of pseudo-cones with given recession cone. We provide a sufficient condition for possibly infinite measures and a special class of pseudo-cones.

A reverse Minkowski theorem

A reverse Minkowski theorem

Proof of the Brunn–Minkowski Theorem by Elementary Methods

Proof of the Brunn–Minkowski Theorem by Elementary Methods

Proof of the Brunn–Minkowski Theorem by Brunn Cuts

Proof of the Brunn–Minkowski Theorem by Brunn Cuts

Asymptotic analysis of spin-foams with timelike faces in a new parametrization

In this article we study the Conrady-Hnybida extension of the Lorentzian Engle-Pereira-Rovelli-Livine spin-foam model, which admits timelike cells rather than just spacelike ones. Our focus is on the asymptotic analysis of the model's vertex amplitude. We propose a new parametrization for states associated to timelike 3-cells, from which we derive a closed-form expression for their amplitudes. This allows us to revisit the conditions under which critical points of the amplitudes occur, and we find Regge-like geometrical critical points in agreement with the literature. However, we find also evidence for nongeometrical points which are not dynamically suppressed without further assumptions; the model then does not strictly asymptote to the Regge action, contrary to what one would expect. We moreover prove Minkowski and rigidity theorems for Minkowskian polyhedra, extending the asymptotic analysis to nonsimplicial spin-foams.

Dilation theory in finite dimensions and matrix convexity

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carath\'{e}odory's and of Minkowski's theorem for matrix convex sets.

Mass transference principle from rectangles to rectangles in Diophantine approximation

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity. The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Minkowski's theorem. So the result sets up a general principle for the Hausdorff measure theory for high dimensional Diophantine approximation which, together with the landmark work of Beresnevich & Velani in 2006 where a transference principle from balls to balls is established, gives a coherent Hausdorff measure theory for metric Diophantine approximation. The dimensional theory for limsup sets generated by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods or even their generalizations fail to work.

Ciphertext-Only Attack on RSA Using Lattice Basis Reduction

We use lattice basis reduction for ciphertext-only attack on RSA. Our attack is applicable in the conditions when known attacks are not applicable, and, contrary to known attacks, it does not require prior knowledge of a part of a message or key, small encryption key, e, or message broadcasting. Our attack is successful when a vector, comprised of a message and its exponent, is likely to be the shortest in the lattice, and meets Minkowski's Second Theorem bound. We have conducted experiments for message, keys, and encryption/decryption keys with sizes from 40 to 8193 bits, with dozens of thousands of successful RSA cracks. It took about 45 seconds for cracking 2001 messages of 2050 bits and for large public key values related with Euler’s totient function, and the same order private keys. Based on our findings, for RSA not to be susceptible to the proposed attack, it is recommended avoiding RSA public key form used in our experiments

Retracted: The Hasse-Minkowski Theorem and Legendre's Theorem for Quadratic Forms in Two and Three Variables

Determining the solvability of equations has been an extended and fundamental study in Mathematics. The local-global principle states two objects are equivalent globally if and only if they are equivalent locally at all places. By applying this principle, the Hasse - Minkowski theorem is able to identify the existence of rational solutions of an equation. This paper explores the applications of the Hasse-Minkowski theorem to homogeneous quadratic forms in two and three variables. After providing some of the necessary proofs and definitions, we have been able to introduce some complete computer programs implementing the Hasse-Minkowski theorems and Legendre theorem with some supporting functions like the Eratosthenes sieve.This article was retracted on January 03, 2022.

Inhomogeneous Diophantine approximation over fields of formal power series

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.

An automorphic generalization of the Hermite–Minkowski theorem

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.

Complete and computable orbit invariants in the geometry of the affine group over the integers

The subject matter of this paper is the geometry of the affine group over the integers, $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ . Turing-computable complete $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch–Jung continued fraction algorithm. We then consider rational polyhedra, i.e., finite unions of simplexes in $${\mathbb {R}}^n$$ with rational vertices. Markov’s unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra P and $$P'$$ are continuously $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -equidissectable. The same problem for the continuous $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -equidissectability of P and $$P'$$ is open. We prove the decidability of the problem whether two rational polyhedra P and $$P'$$ in $${\mathbb {R}}^n$$ have the same $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit.

An extension of Minkowski's theorem and its applications to questions about projections for measures

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and the surface area measure determines a convex body uniquely up to a shift. In this manuscript we prove an extension of Minkowski's theorem. Consider a measure μ on Rn with positive degree of concavity and positive degree of homogeneity. We show that a surface area measure of a convex set K, weighted with respect to μ, determines a convex body uniquely up to μ-measure zero. We also establish an existence result under natural conditions including symmetry.We apply this result to extend the solution to classical Shephard's problem. To do this, we introduce a new notion which relates projections of convex bodies to a given measure μ, and is a generalization of the Lebesgue volume of a projection.

Faster integer multiplication using short lattice vectors

We prove that $n$-bit integers may be multiplied in $O(n \log n \, 4^{\log^* n})$ bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is unconditional, and depends in an essential way on Minkowski's theorem concerning lattice vectors in symmetric convex sets.

Reductions in PPP

We show several reductions between problems in the complexity class PPP related to the pigeonhole principle, and harboring several intriguing problems relevant to Cryptography. We define a problem related to Minkowski's theorem and another related to Dirichlet's theorem, and we show them to belong to this class. We also show that Minkowski is very expressive, in the sense that all other non-generic problems in PPP considered here can be reduced to it. We conjecture that Minkowski is PPP-complete.

Local Okounkov bodies and limits in prime characteristic

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert–Kunz multiplicity, and F-signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton–Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert–Kunz multiplicity and F-signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn–Minkowski theorem for Hilbert–Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.

On common zeros of a pair of quadratic forms over a finite field

Let F be a finite field of characteristic distinct from 2, f and g quadratic forms over F, dim⁡f=dim⁡g=n. A particular case of Chevalley's theorem claims that if n≥5, then f and g have a common zero. We give an algorithm, which establishes whether f and g have a common zero in the case n≤4. The most interesting case is n=4. In particular, we show that if n=4 and det⁡(f+tg) is a squarefree polynomial of degree different from 2, then f and g have a common zero. We investigate the orbits of pairs of 4-dimensional forms (f,g) under the action of the group GL4(F), provided f and g do not have a common zero. In particular, it turns out that for any polynomial p(t) of degree at most 4 up to the above action there exist at most two pairs (f,g) such that det⁡(f+tg)=p(t), and the forms f, g do not have a common zero. The proofs heavily use Brumer's theorem and the Hasse–Minkowski theorem.

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.

Lattices and Rational Points

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N - 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q -rank ≤ N - 1 . We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author.