scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

Image Milnor number and 𝒜e-codimension for maps between weighted homogeneous irreducible curves

2020 ◽  
Vol 20 (3) ◽  
pp. 319-330
Author(s):  
D. A. H. Ament ◽  
J. J. Nuño-Ballesteros ◽  
J. N. Tomazella
Keyword(s):  
Milnor Number ◽  
Versal Deformation ◽  
Minimum Number ◽  
Vanishing Cycles ◽  
One To One

AbstractLet (X, 0) ⊂ (ℂn, 0) be an irreducible weighted homogeneous singularity curve and let f : (X, 0) → (ℂ2, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (X, 0). We show that 𝒜e-codim(X, f) = μI(f), where the 𝒜e-codimension 𝒜e-codim(X, f) is the minimum number of parameters in a versal deformation and μI(f) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of f.

scholarly journals Méthodes fonctionnelles pour la transcendance en caractéristique finie

1994 ◽  
Vol 50 (2) ◽  
pp. 273-286 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Exponential Function ◽  
Characteristic Zero ◽  
Algebraic Closure ◽  
Algebraic Groups ◽  
Zero Element ◽  
Finite Characteristic ◽  
First Derivative ◽  
Linearly Independent ◽  
Divisibility Properties

There are essentially two ways to obtain transcendence results in finite characteristic. The first, historically, is to use Ore's lemma and to prove that a series whose coefficients satisfy well-behaved divisibility properties cannot be a zero of an additive polynomial. This method is of the same kind as the method of p–automata. The second one is to try to imitate the usual methods in characteristic zero and to do transcendence theory with t–modules analogously to what we can do with algebraic groups. We want to show here that transcendence results over Fq(T) can also be obtained with the help of the variable T. If ec(z) is the Carlitz exponential function and e = ec(1), we obtain, in particular, that 1, e, …, e(p–2) (the P–2 first derivative of e with respect to T) are linearly independent over the algebraic closure of Fq(T). A corollary is that for every non-zero element α in Fq((1/T)), αpe and αec(e1/p) are transcendental over Fq(T). By changing the variable and using older results we also obtain the transcendence of ec(ω) for all ω ∈ Fq((1/T)) such that ω(T) and ω(Ti) are not zero and linearly dependent over Fq (Ti) (q > 2i + 1). Such u appear to be transcendental by the method of Mahler if i is not a power of p.

scholarly journals Beck's Theorem for Plane Curves

10.37236/9658 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Mario Huicochea
Keyword(s):  
Finite Subset ◽  
Characteristic Zero ◽  
Algebraic Curves ◽  
Plane Curves ◽  
The Family

 Let $d\in\mathbb{Z}^+t$, $\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\mathbb{K}^2$. Denote by $\mathcal{C}_{d,\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\mathbb{K}^2$ and $\mathcal{C}_{\leq d,\mathbb{K}}:=\bigcup_{e=1}^d\mathcal{C}_{e,\mathbb{K}}$. For any $C_1\in \mathcal{C}_{d,\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\in\mathcal{C}{d,\mathbb{K}}$ such that $C_2\cap A\supseteq C_1\cap A$, we have that $C_1=C_2$; we denote by $\mathcal{D}_{d,\mathbb{K}}(A)$ the family of elements of $\mathcal{C}_{d,\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\mathbb{K}=\mathbb{R}$ and $A$ is not collinear, then $$|\mathcal{D}_{1,\mathbb{R}}(A)|=\Theta\left(|A|\min_{C\in \mathcal{C}_{1,\mathbb{R}}}|A\setminus C|\right).$$ In this paper we generalize Beck's theorem showing that for all $d\in\mathbb{Z}^+$, there exists a constant $c=c(d)>0$ such that if  $\min_{C\in\mathcal{C}_{\leq d,\mathbb{K}}}|A\setminus C|>c,$ then  $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$

Cartan Subalgebras of Zassenhaus Algebras

1975 ◽  
Vol 27 (5) ◽  
pp. 1011-1021 ◽  
Author(s):  
Gordon Brown
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Complete Classification ◽  
The Other ◽  
Closed Field ◽  
Finite Characteristic ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Cartan Subalgebras

Cartan subalgebras play a very important role in the classification of the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero. It is well-known [5, 273] that any two Cartan subalgebras of such an algebra are conjugate, i.e. images of one another under some automorphism of the algebra. On the other hand, there exist finitedimensional simple Lie algebras over fields of finite characteristic p possessing non-conjugate Cartan subalgebras [2; 3; 4]. The simple Lie algebras discovered by Zassenhaus [6] also possess non-conjugate Cartan subalgebras, and we shall give a complete classification of Cartan subalgebras of these algebras in this paper.

Kinematic analysis and deformation of a planar lattice with an arbitrary number of panels

2020 ◽  
pp. 15-19
Author(s):  
M.N. Kirsanov
Keyword(s):  
Exact Solution ◽  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Design Parameters ◽  
Lattice Deformation ◽  
Planar Lattice ◽  
Force Loading

Formulae are obtained for calculating the deformations of a statically determinate lattice under the action of two types of loads in its plane, depending on the number of panels located along one side of the lattice. Two options for fixing the lattice are analyzed. Cases of kinematic variability of the structure are found. The distribution of forces in the rods of the lattice is shown. The dependences of the force loading of some rods on the design parameters are obtained. Keywords: truss, lattice, deformation, exact solution, deflection, induction, Maple system. [email protected]

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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