Intereting Posts

Drunkards walk on a sphere.
How to efficiently generate five numbers that add to one?
$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$
Axiom of Choice: Can someone explain the fallacy in this reasoning?
Pullback metric, coordinate vector fields..
Is the equation $\phi(\pi(\phi^\pi)) = 1$ true? And if so, how?
Difference between $\mathbb C$ and $\mathbb R^2$
Spaces with equal homotopy groups but different homology groups?
conjectured general continued fraction for the quotient of gamma functions
Sobolev spaces of sections of vector bundles
Complex Numbers Geometry
Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?
Mean distance between 2 points within a sphere
Origin of the terminology projective module

Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points?

In view of the Jacobian test for regularity (which is just the implicit function theorem in this case), it suffices to show that there exist a point $x\in V$ and polynomials $f_1,…,f_p$ in the defining ideal $I$ of $V$ so that the derivatives $df_1,…, df_p$ are linearly independent at $x$. However, I do not see why such point and polynomials would exist.

- Ring germs of $C^{\infty}$ functions on the real line
- Sheaf cohomology: what is it and where can I learn it?
- Cancellation problem: $R\not\cong S$ but $R\cong S$ (Danielewski surfaces)
- Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
- Connections of Geometric Group Theory with other areas of mathematics.
- Monic (epi) natural transformations

- Why can't the Polynomial Ring be a Field?
- In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
- How to determine a set of polynomials is algebraically indepedent or not?
- The Picard-Brauer short exact sequence
- Singular affine real varieties are no manifolds?
- Requirements on ring for injective-projectiveness
- Quotient of polynomials, PID but not Euclidean domain?
- Is the intersection of two f.g. projective submodules f.g.?
- When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof
- Showing an ideal is prime in polynomial ring

- Does $\lfloor(4+\sqrt{11})^{n}\rfloor \pmod {100}$ repeat every $20$ cycles of $n$?
- Lambert Function as a solution
- Eigenvalues of $A+B$
- Clopen subsets of a compact metric space
- Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.
- If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
- Trying to understand the use of the “word” pullback/pushforward.
- Understanding of exterior algebra
- integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$
- Describe all one-dimensional representations of the alternating group A4.
- Calculating the limit of $^{1/n}$ as $n$ tends to $\infty$
- Apparent inconsistency between integral table and integration using trigonometric identity
- Moment generating function and exponentially decaying tails of probability distribution
- Disjoint Refinement
- Estimating the number of integers less than $m$ that are relatively prime to $p_n\#$