scholarly journals On Catalan Trees and the Jacobian Conjecture

10.37236/1546 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Dan Singer
Keyword(s):  
Equivalence Classes ◽  
Jacobian Conjecture ◽  
Complex Numbers ◽  
Total Degree ◽  
Combinatorial Properties ◽  
Algebraic Result ◽  
Ring Of Polynomials

New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let $F=(x_1+H_1,x_2+H_2,\dots,x_n+H_n)$ be a system of $n$ polynomials in $C[x_1,x_2,\dots,x_n]$, the ring of polynomials in the variables $x_1,x_2, \dots, x_n$ over the field of complex numbers. Let $H=(H_1,H_2,\dots,H_n)$. Our principal algebraic result is that if the Jacobian of $F$ is equal to 1, the polynomials $H_i$ are each homogeneous of total degree 2, and $({{\partial H_i}\over {\partial x_j}})^3=0$, then $H\circ H\circ H=0$ and $F$ has an inverse of the form $G=(G_1,G_2,\dots,G_n)$, where each $G_i$ is a polynomial of total degree $\le6$. We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero. We also show that if all of the polynomials $H_i$ are homogeneous of the same total degree $d\ge2$ and $({{\partial H_i}\over {\partial x_j}})^2=0$, then $H\circ H=0$ and the inverse of $F$ is $G=(x_1-H_1,\dots,x_n-H_n)$.

scholarly journals Completions of rings of invariants and their divisor class groups

1978 ◽  
Vol 72 ◽  
pp. 71-82 ◽  
Author(s):  
Phillip Griffith
Keyword(s):  
Power Series ◽  
Maximal Ideal ◽  
Total Degree ◽  
Class Groups ◽  
Divisor Class ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Ring Of Polynomials ◽  
Formal Power

Let k be a field and let A = be a normal graded subring of the full ring of polynomials R = k[X1, · · ·, Xn] (where R always is graded via total degree and A0 = k). R. Fossum and the author [F-G] observed that the completion  at the irrelevant maximal ideal of A is isomorphic to the subring of the formal power series ring R̂ = k[[X1, · ·., Xn]] and, moreover, that  is a ring of invariants of an algebraic group whenever A is.

Moduli of Rank 2 Stable Bundles and Hecke Curves

2016 ◽  
Vol 59 (4) ◽  
pp. 865-877
Author(s):  
Sarbeswar Pal
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Short Note ◽  
Equivalence Classes ◽  
Complex Numbers ◽  
Stable Bundles ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Arbitrary Genus ◽  
Rank 2

AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

scholarly journals The bordism group of unbounded KK-cycles

2018 ◽  
Vol 10 (02) ◽  
pp. 355-400 ◽  
Author(s):  
Robin J. Deeley ◽  
Magnus Goffeng ◽  
Bram Mesland
Keyword(s):  
Abelian Group ◽  
Equivalence Relation ◽  
Compact Manifold ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Lipschitz Functions ◽  
Complex Numbers ◽  
Manifold With Boundary ◽  
Dense Subalgebra ◽  
Bordism Group

We consider Hilsum’s notion of bordism as an equivalence relation on unbounded [Formula: see text]-cycles and study the equivalence classes. Upon fixing two [Formula: see text]-algebras, and a ∗-subalgebra dense in the first [Formula: see text]-algebra, a [Formula: see text]-graded abelian group is obtained; it maps to the Kasparov [Formula: see text]-group of the two [Formula: see text]-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first [Formula: see text]-algebra is the complex numbers (i.e. for [Formula: see text]-theory) and is a split surjection if the first [Formula: see text]-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense ∗-subalgebra.

scholarly journals On the Combinatorial Structure of Arrangements of Oriented Pseudocircles

10.37236/1783 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Johann Linhart ◽  
Ronald Ortner
Keyword(s):  
Representation Theorem ◽  
Equivalence Classes ◽  
Oriented Matroids ◽  
Combinatorial Structure ◽  
Topological Representation ◽  
Combinatorial Properties ◽  
One To One ◽  
Orientable Surfaces

We introduce intersection schemes (a generalization of uniform oriented matroids of rank 3) to describe the combinatorial properties of arrangements of pseudocircles in the plane and on closed orientable surfaces. Similar to the Folkman-Lawrence topological representation theorem for oriented matroids we show that there is a one-to-one correspondence between intersection schemes and equivalence classes of arrangements of pseudocircles. Furthermore, we consider arrangements where the pseudocircles separate the surface into two components. For these strict arrangements there is a one-to-one correspondence to a quite natural subclass of consistent intersection schemes.

scholarly journals Toward a Combinatorial Proof of the Jacobian Conjecture!

10.37236/2023 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Dan Singer
Keyword(s):  
Vector Space ◽  
Full Rank ◽  
Coefficient Matrix ◽  
Rooted Trees ◽  
Jacobian Conjecture ◽  
Combinatorial Proof ◽  
Combinatorial Properties ◽  
Linearly Independent ◽  
Acyclic Digraph ◽  
Fixed Tree

The Jacobian conjecture [Keller, Monatsh. Math. Phys., 1939] gives rise to a problem in combinatorial linear algebra: Is the vector space generated by rooted trees spanned by forest shuffle vectors? In order to make headway on this problem we must study the algebraic and combinatorial properties of rooted trees. We prove three theorems about the vector space generated by binary rooted trees: Shuffle vectors of fixed length forests are linearly independent, shuffle vectors of nondegenerate forests relative to a fixed tree are linearly independent, and shuffle vectors of sufficient length forests are linearly independent. These results are proved using the acyclic digraph method for establishing that a coefficient matrix has full rank [Singer, The Electronic Journal of Combinatorics, 2009]. We also provide an infinite class of counterexamples to demonstrate the need for sufficient length in the third theorem.

An ω1-categorical ring which is not almost strongly minimal

1974 ◽  
Vol 39 (4) ◽  
pp. 665-668 ◽  
Author(s):  
K.-P. Podewski ◽  
J. Reineke
Keyword(s):  
Commutative Ring ◽  
Order Logic ◽  
First Order Logic ◽  
Complex Numbers ◽  
First Order ◽  
Factor Ring ◽  
Closed Fields ◽  
Ring Of Polynomials ◽  
Algebraically Closed Fields ◽  
The Ideal

A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].It is straightforward to prove that R0 has the following properties:1. R0 is a commutative ring with identity.2. R0 is of characteristic 0.3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.5. There is an x ≠ 0 such that x2 = 0.These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.

scholarly journals Module parallel transports in fuzzy gauge theory

2014 ◽  
Vol 11 (03) ◽  
pp. 1450021
Author(s):  
Alexander Schenkel
Keyword(s):  
Parallel Transport ◽  
Equivalence Classes ◽  
Complex Numbers ◽  
Projective Modules ◽  
Gauge Invariant ◽  
Matrix Algebras ◽  
Gauge Equivalence ◽  
Finite Matrix ◽  
Basic Set ◽  
The One

In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

scholarly journals On the Bateman–Horn conjecture for polynomials over large finite fields

2016 ◽  
Vol 152 (12) ◽  
pp. 2525-2544 ◽  
Author(s):  
Alexei Entin
Keyword(s):  
Finite Field ◽  
Affine Plane ◽  
Plane Curve ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Total Degree ◽  
Number Of Factors ◽  
Ring Of Polynomials ◽  
Image Position

We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable$x$) polynomials$F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$, we show that the number of$f\in \mathbf{F}_{q}[t]$of degree$n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$such that all$F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$, are irreducible is$$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$where$N_{i}=n\deg _{x}F_{i}$is the generic degree of$F_{i}(t,f)$for$\deg f=n$and$\unicode[STIX]{x1D707}_{i}$is the number of factors into which$F_{i}$splits over$\overline{\mathbf{F}}_{q}$. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over$\mathbf{F}_{q}(t)$) polynomials$F_{1},\ldots ,F_{m}$not necessarily monic in$x$under the assumptions that$n$is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve$C$defined by the equation$$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{m}F_{i}(t,x)=0 & & \displaystyle \nonumber\end{eqnarray}$$(this number is always bounded above by$(\sum _{i=1}^{m}\deg F_{i})^{2}/2$, where$\deg$denotes the total degree in$t,x$) and$$\begin{eqnarray}\displaystyle p=\text{char}\,\mathbf{F}_{q}>\max _{1\leqslant i\leqslant m}N_{i}, & & \displaystyle \nonumber\end{eqnarray}$$where$N_{i}$is the generic degree of$F_{i}(t,f)$for$\deg f=n$.

Complex numbers

2006 ◽  
Author(s):  
Stephen C. Roy
Keyword(s):  
Complex Numbers
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