Intereting Posts

GRE problem involving LCD, prime factorization, and sets.
How was Euler able to create an infinite product for sinc by using its roots?
How can I find equivalent Euler angles?
Prove that limit $\lim_{n\to\infty}\sqrt{4+\frac{1}{n^2}}+\sqrt{4+\frac{2}{n^2}}+\cdots+ \sqrt{4+\frac{n}{n^2}}-2n=\frac{1}{8}$
Conjecture regarding integrals of the form $\int_0^\infty \frac{(\log{x})^n}{1+x^2}\,\mathrm{d}x$.
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized
Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$
Is empty set element of every set if it is subset of every set?
Difficult limit problem involving sine and tangent
Prove this inequality with $xyz\le 1$
Definition of independence of infinite random variables
Making an infinite generating function a finite one
Show an exponential function has a valid density.
noether normalization theorem geometric meaning
Finding the fraction $\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}$ when knowing the sums $a+b+c+d$ to $a^4+b^4+c^4+d^4$

Given an irreducible polynomial $P(X,Y) \in \mathbb{C}[X,Y]$, one obtains by analytic continuation a Riemann surface $M$ with a branched covering $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, corresponding to $P(z,f) = 0$.

Of particular interest in this setup is the monodromy, i.e. the action of the fundamental group of $\mathbb{P}^1(\mathbb{C}) \setminus B$ (where $B$ is the set of branch points) on a fiber above some generic point $p$.

To compute the relevant local monodromies (around each individual branch point), one can compute Puiseux series around the branch point and read off the monodromy data.

- Does S-equivalence imply a deformation relation?
- Applications of Belyi's theorem
- Tangent sheaf of a (specific) nodal curve
- How can I get smooth curve at the sigmoid function?
- For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .
- Solving the curve equation for logarithmic decay using two anchor points.

How exactly does one compute, given only the polynomial $P(X,Y)$, what the different Puiseux series look like? Is there a more intuitive way of seeing what the cover looks like without delving into lengthy calculations with power series?

- Complex analysis book with a view toward Riemann surfaces?
- Calculating the projective closure with more than one generator
- What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?
- Can $\mathbb RP^2$ cover $\mathbb S^2$?
- Valuation rings and total order
- Applications of Belyi's theorem
- Number of fixed points of automorphism on Riemann Surface
- When do equations represent the same curve?
- Why $y=e^x$ is not an algebraic curve?
- Is the derivative of a modular function a modular function

The potential ramification points are $X_0$ s.t. either $P$ and $\partial P/\partial Y$ have a common zero at some $(X_0,Y_0)$, or the degree of $P(X_0,Y)\in\mathbb{C}[Y]$ is less than $n$, or $X_0=\infty$. Let us consider the first case, i.e. suppose that $Y_0\in\mathbb{C}$ is a $k$-fold root of $P(X_0,Y)\in\mathbb{C}[Y]$ (the remaining cases are similar). For $X_1$ close to $X_0$ this multiple root will split into $k$ different roots. Generically we will get a $k$-cycle in the local monodromy at $X_0$ (the roots undergo a cyclic permutation when we go around $X_0$). Sometimes we may, however, get a different permutation of these $k$ roots. It can be determined using Newton polygon as follows.

We may suppose that $(X_0,Y_0)=(0,0)$ (by shifting the variables).

For every monomial $Y^aX^b$ in $P$ we draw the point $(a,b)$ in the plane and then we take the convex hull of these points. We consider only the part of the boundary of the resulting polygon from which $(0,0)$ is visible. Let $s_1,\dots,s_q$ be these sides, and let $k_1,\dots,k_q$ be the lengths of the projections of $s_i$’s to the $x$-axis; we clearly have $k_1+\dots+k_q=k$. If none of the sides $s_i$’s contain any integer points inside then $(k_1,\dots,k_q)$ are the lengths of the cycles of the local monodromy.

For example, for the polynomial $X^3+ 5XY^3-8Y^7+iY^{10}+(2+i)X^4-X^2Y^4$ we get $-8Y^7+iY^{10}$ when we set $X=0$, i.e. $Y=0$ is a $7$-fold root, and the Newton polygon is

so that we get a $3$-cycle and a $4$-cycle.

(If some of $s_i$ contain an integer point inside then things become more complicated: essentially we know that we have a Puiseux series solution $Y=cX^t+\dots$ of $P(X,Y)=0$, where $-t$ is the slope of $s_i$, and we need to determine what denominators appear in the exponents of this series. We basically pose $Z=Y-cX^t$, get an equation for $Z$, and use again Newton polygon to see what happens.)

Newton polygon is equally useful for determining ramifications for finite extensions of $p$-adic numbers (the method is the same).

- Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
- Least upper bound property implies Cauchy completeness
- How to apply the recursion theorem in practice?
- Tips for understanding the unit circle
- To show that group G is abelian if $(ab)^3 = a^3 b^3$ and the order of $G$ is not divisible by 3
- How to prove that $x^a-1|x^b-1 \Longleftrightarrow a|b$.
- A necessary condition for series convergence with positive monotonically decreasing terms
- What is the minimum number of locks on the cabinet that would satisfy these conditions?
- The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$
- Ramanujan 691 congruence
- Inhomogeneous equation
- Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.
- Why $e^x$ is always greater than $x^e$?
- What should the high school math curriculum consist of?
- Spectrum of difference of two projections