Transcendence Degree Research Articles

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

The general problem studied in this paper is to decompose the positive elements R+ of a totally ordered ring R into the disjoint union of sets that are closed under addition and multiplication. We mainly investigate the case when R is a subfield of the reals R. We prove that for any n∈N the positive real numbers can be decomposed into n disjoint pieces which are also closed under addition and multiplication. We construct infinitely many different n-decompositions (n∈N) for all fields containing at least two algebraically independent transcendental numbers. We characterise all possible decompositions into two pieces for fields of transcendence degree one over Q. Further, we prove that the positive elements of real algebraic extensions of the rational numbers are indecomposable into finitely many pieces.

On the representation of fields as sums of two proper subfields

In this paper, we study which field F can be represented as sums of two proper subfields. We prove that a field F is a sum of two proper subfields if and only if the transcendence degree of F over its prime subfield is infinite. This answers a question posed in the paper [M. Kȩpczyk and R. Mazurek, On the representation of fields as finite sums of proper subfields, Results Math. 75(2) (2020) 65].

Abelian absolute Galois groups

AbstractGeneralizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $\textrm{Gal}(K)$ , then ${\mathrm{rank}}(A)\le r+1$ . Moreover, if $\mathrm{char}(K)=0$ , then ${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $\textrm{Gal}(K)$ .

Strongly dense free subgroups of semisimple algebraic groups II

It was shown in [9] that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least 1 in characteristic 0 and at least 2 in positive characteristic. We also consider related questions for surface groups.

Splitting of differential quaternion algebras

We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field k is always split by a quadratic extension of k. However, a differential quaternion algebra need not be split over any algebraic extension of k. We use solutions of certain Riccati equations to provide bounds on the transcendence degree of splitting fields of a differential quaternion algebra.

A 4-fold categorical equivalence

In this note, we will illuminate some immediate consequences of work done by Reineke in [Algebr. Represent. Theory 16 (2013), no. 5. 1313–1314] that may prove to be useful in the study of elliptic curves. In particular, we will construct an isomorphism between the category of smooth projective curves with a category of quiver Grassmannians. We will use this to provide a 4-fold categorical equivalence between a category of quiver Grassmannians, smooth projective curves, compact Riemann surfaces, and fields of transcendence degree 1 over C \mathbb {C} . We finish with noting that the category of elliptic curves is isomorphic to a category of quiver Grassmannians, whence providing an analytic group structure to a class of quiver Grassmannians.

Isomorphism between the Algebra of Measurable Functions and its Subalgebra of Approximately Differentiable Functions

The present paper is devoted to study of certain classes of homogeneous regular subalgebras of the algebra of all complex-valued measurable functions on the unit interval. It is known that the transcendence degree of a commutative unital regular algebra is one of the important invariants of such algebras together with Boolean algebra of its idempotents. It is also known that if $(\Omega, \Sigma, \mu)$ is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of $S(\Omega)$ are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and transcendence degrees of these algebras coincide. Let $S(0,1)$ be the algebra of all (classes of equivalence) measurable complex-valued functions and let $AD^{(n)}(0,1)$ ($n\in \mathbb{N}\cup\{\infty\}$) be the algebra of all (classes of equivalence of) almost everywhere $n$-times approximately differentiable functions on $[0,1].$ We prove that $AD^{(n)}(0,1)$ is a regular, integrally closed, $\rho$-closed, $c$-homogeneous subalgebra in $S(0,1)$ for all $n\in \mathbb{N}\cup\{\infty\},$ where $c$ is the continuum. Further we show that the algebras $S(0,1)$ and $AD^{(n)}(0,1)$ are isomorphic for all $n\in \mathbb{N}\cup\{\infty\}.$ As an application of these results we obtain that the dimension of the linear space of all derivations on $S(0,1)$ and the order of the group of all band preserving automorphisms of $S(0,1)$ coincide and are equal to $2^c.$ Finally, we show that the Lie algebra $\operatorname{Der} S(0, 1)$ of all derivations on $S(0,1)$ contains a subalgebra isomorphic to the infinite dimensional Witt algebra.

Hay from the Haystack: Explicit Examples of Exponential Quantum Circuit Complexity

The vast majority of quantum states and unitaries have circuit complexity exponential in the number of qubits. In a similar vein, most of them also have exponential minimum description length, which makes it difficult to pinpoint examples of exponential complexity. In this work, we construct examples of constant description length but exponential circuit complexity. We provide infinite families such that each element requires an exponential number of two-qubit gates to be generated exactly from a product and where the same is true for the approximate generation of the vast majority of elements in the family. The results are based on sets of large transcendence degree and discussed for tensor networks, diagonal unitaries and maximally coherent states.

Hasse principles for quadratic forms over function fields

We investigate the Hasse principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Over purely transcendental field extensions of fields that satisfy property Ai(2) for some i, we find numerous counterexamples to the Hasse principle for isotropy with respect to a relatively small set of discrete valuations. For finitely generated field extensions K of transcendence degree r over an algebraically closed field of characteristic ≠2, we use the 2r-dimensional counterexample to the Hasse principle for isotropy due to Auel and Suresh to obtain counterexamples of lower dimensions with respect to the divisorial discrete valuations induced by a variety with function field K.

Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [ 3 ] and exploited by them, and Agrawal et al. [ 2 ] to design algebraic independence–based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey et al. [ 15 ], we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. [ 3 ] and Agrawal et al. [ 2 ] in the positive characteristic setting.

Bi-Condition of Existence for a Compatible Directed Order on an Arbitrary Field

One proves that a field carries a compatible directed order if and only if it has characteristic 0 and is real or has a transcendence degree of at least 1 over the field of rational numbers.

Subgroups of free proalgebraic groups and Matzat’s conjecture for function fields

We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. As an application, we prove that the absolute differential Galois group of a one-variable function field over an algebraically closed characteristic zero field of infinite transcendence degree is a free proalgebraic group.

Splitting fields of differential symbol algebras

For m≥2, we study derivations on symbol algebras of degree m over fields with characteristic not dividing m. A differential central simple algebra over a field k is split by a finitely generated extension of k. For certain derivations on symbol algebras, we provide explicit construction of differential splitting fields and give bounds on their algebraic and transcendence degrees. We further analyze maximal subfields that split certain differential symbol algebras.

Local cohomology bounds and the weak implies strong conjecture in dimension 4

We verify the weak implies strong conjecture for 4-dimensional finitely generated algebras over a field of prime characteristic p>5 which has infinite transcendence degree over Fp.

The existence of Zariski dense orbits for endomorphisms of projective surfaces

Let f f be a dominant endomorphism of a smooth projective surface X X over an algebraically closed field k \mathbf {k} of characteristic 0 0 . We prove that if there is no rational function H ∈ k ( X ) H \in \mathbf {k}(X) such that H ∘ f = H H \circ f = H , then there exists a point x ∈ X ( k ) x \in X(\mathbf {k}) such that the forward orbit of x x under f f is Zariski dense in X X . This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of smooth projective surfaces. We also define a new topology on varieties over algebraically closed fields with finite transcendence degree over Q \mathbb {Q} , which we call “the adelic topology”. This topology is stronger than the Zariski topology, and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic version of the Zariski dense orbit conjecture that is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of smooth projective surfaces, for endomorphisms of abelian varieties, and for split polynomial maps. This yields new proofs of the original conjecture in the latter two cases. In Appendix A, we study endomorphisms of k k -affinoid spaces. We show that for certain endomorphisms f f on a k k -affinoid space X X , the attractor Y Y of f f is a Zariski closed subset and that the dynamics of f f is semi-conjugate to its restriction to Y Y . A special case of this result is used in the proof of the main theorem. In Appendix B, written in collaboration with Thomas Tucker, we prove the Zariski dense orbit conjecture for endomorphisms of ( P 1 ) N (\mathbb {P}^1)^N .

A simple and constructive proof to a generalization of Lüroth's theorem

A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this generalization.

Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery model

AbstractWe study the nonconvex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This is a statistical model motivated by applications in molecular microscopy and image processing, where each measurement of an unknown object is subject to an independent random rotation from a known rotational group. Equivalently, it is a Gaussian mixture model where the mixture centers belong to a group orbit.We show that fundamental properties of the likelihood landscape depend on the signal‐to‐noise ratio and the group structure. At low noise, this landscape is “benign” for any discrete group, possessing no spurious local optima and only strict saddle points. At high noise, this landscape may develop spurious local optima, depending on the specific group. We discuss several positive and negative examples, and provide a general condition that ensures a globally benign landscape at high noise. For cyclic permutations of coordinates on (multireference alignment), there may be spurious local optima when , and we establish a correspondence between these local optima and those of a surrogate function of the phase variables in the Fourier domain.We show that the Fisher information matrix transitions from resembling that of a single Gaussian distribution in low noise to having a graded eigenvalue structure in high noise, which is determined by the graded algebra of invariant polynomials under the group action. In a local neighborhood of the true object, where the neighborhood size is independent of the signal‐to‐noise ratio, the landscape is strongly convex in a reparametrized system of variables given by a transcendence basis of this polynomial algebra. We discuss implications for optimization algorithms, including slow convergence of expectation‐maximization, and possible advantages of momentum‐based acceleration and variable reparametrization for first‐ and second‐order descent methods. © 2021 Wiley Periodicals LLC.

Some co-tame automorphisms of affine spaces

The investigation of co-tame automorphisms of the affine space [Formula: see text] is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.

Observer Based Multi-Level Fault Reconstruction for Interconnected Systems.

The problem of local fault (unknown input) reconstruction for interconnected systems is addressed in this paper. This contribution consists of a geometric method which solves the fault reconstruction (FR) problem via observer based and a differential algebraic concept. The fault diagnosis (FD) problem is tackled using the concept of the differential transcendence degree of a differential field extension and the algebraic observability. The goal is to examine whether the fault occurring in the low-level subsystem can be reconstructed correctly by the output at the high-level subsystem under given initial states. By introducing the fault as an additional state of the low subsystem, an observer based approached is proposed to estimate this new state. Particularly, the output of the lower subsystem is assumed unknown, and is considered as auxiliary outputs. Then, the auxiliary outputs are estimated by a sliding mode observer which is generated by using global outputs and inverse techniques. After this, the estimated auxiliary outputs are employed as virtual sensors of the system to generate a reduced-order observer, which is caplable of estimating the fault variable asymptotically. Thus, the purpose of multi-level fault reconstruction is achieved. Numerical simulations on an intensified heat exchanger are presented to illustrate the effectiveness of the proposed approach.

Some Automorphism Groups are Linear Algebraic

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over $k$, then Aut($X$) is an algebraic group.