8
votes
1answer
179 views

Differential analogue of the newton polygon

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows: Let us recall the ...
1
vote
0answers
79 views

G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...
2
votes
0answers
239 views

A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } ...
3
votes
2answers
256 views

existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference. Let $k$ be some (algebraically closed, if needed) field. There is a ...
1
vote
1answer
123 views

Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...
17
votes
0answers
845 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
1
vote
0answers
148 views

“Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone! I have some construction that requires exact definition. I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
1
vote
0answers
151 views

On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest eigenvalue $\mu_{3}$ for the differential equation $\Delta ...
4
votes
4answers
967 views

The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...
8
votes
2answers
617 views

Grothendieck connections and jets

The following question is based on some remarks in section I.2 of Deligne's book Equations Différentielles à Points Singuliers Réguliers. Let $X$ be a smooth complex variety and ...
5
votes
1answer
236 views

D-modules on affine space that are regular at infinity

If I have a D-module $M$ on $\mathbb{A}^n$ (which is essentially the same thing as a module over the Weyl algebra), then I can push this D-module forward to $\mathbb{P}^n$ to get a D-module $j_*M$ on ...
2
votes
0answers
294 views

Boundary behavior of Kähler cone with curvature restriction

Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand. A fundamental result is due to Demailly and Paun: they ...
1
vote
1answer
645 views

Solving nonlinear ODE's

If I would ask for $\phi'(x) = f( \phi(x))$ and $\phi(0)=f(0)$, I would get that the inverse of $\phi$ is forced to be of the form: $$\phi^{-1}(z) = \int_{0}^z \frac{1}{f(x)} d x.$$ Now it is ...
5
votes
1answer
516 views

Exponential sums and differential equations

Hi, I have a general question about the relationship between exponential sums and differential equations. In particular, I have been trying to read Katz' work on the subject (his book and his lecture ...
7
votes
3answers
560 views

In which commutative algebras does any derivation possess a flow?

Definitions Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If ...
7
votes
2answers
1k views

Picard-Fuchs equations for modular functions

Hello, MathOverflow community! Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil ...
5
votes
1answer
563 views

Decide how many non-negative solutions a set of multivariate quadratic equations have

Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have? Some explanations: All the coefficients are real numbers. The number ...
23
votes
1answer
2k views

D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
6
votes
2answers
2k views

Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...