Related papers: On bases that are closed under multiplication
We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple…
Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an…
It was proved that whenever $\mathbb{N}$ is partitioned into finitely many cells, one cell must contain arbitrary length arithmetic and geometric progression nicely intertwined, so that one cell must be rich in the sense of containing…
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
An algebra $\cal{R}$ is called an extension of the algebra $M$ by $B$ if $M^2=0$, $M$ is an ideal of $\cal{R}$ and $\cal{R}$$/M\cong B$ as algebras. In this paper, by using the Gr\"{o}bner-Shirshov bases, we characterize completely the…
For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…
We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…
Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased…
We give necessary and sufficient conditions for the sum of n subspaces of a Hilbert space to be closed. We also present various properties of n-tuples of subspaces with closed sum.
We prove that for an arbitrary $\varepsilon>0$ and any multiplicative subgroup $\Gamma \subseteq \mathbf{F}_p$, $1\ll |\Gamma| \le p^{2/3 -\varepsilon}$ there are no sets $B$, $C \subseteq \mathbf{F}_p$ with $|B|, |C|>1$ such that…
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…
The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper…
We describe families of complete orthogonal bases of full rank matrices which span the vector spaces of square matrices. The proposed bases generalise non-trivially the Pauli matrice while shedding light on their algebraic properties.…
We construct an expansion of a real closed field by a multiplicative subgroup adapting Poizat's theory of green points. Its theory is strongly dependent, and every open set definable in a model of this theory is semialgebraic. We prove that…
We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.
A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in…
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…
In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second…