I would like some help with this question: let $S$ be a sparse subset of $\mathbb {N }$. Let $M$ be a subset of $S$ such that if $m\in M$ and $sa=m$ with $s\in S$ implies that $s=m$ and $a=1$. Let $S(x)$ be the number of represnetations of elements of $S$ less than x. We say that $c(n)$ is the...

Let $K$ be a field, and consider a morphism of locally ringed spaces (or schemes) $f: \operatorname{Spec} K \to Y$. Let $t$ be the unique topological point of $\operatorname{Spec} K$, and suppose that $f(t) \in V$, where $V$ is an affine open subset of $Y$. My question is: Can $f$ be uniquely...

I am working through exercise 14.1.B(c) in Ravi Vakil's excellent Fundamentals of Algebraic Geometry. I'll here reproduce the statement: Exercise 14.1.B(c) - Let $\pi:X \to Y$ be a morphism of ringed spaces. If $\mathscr G$ is a locally free sheaf of rank $n$ on $Y$ and $\{U_i\}$ are...

Fix a commutative ring $k$. $\def\Spec{\operatorname{Spec}}\def\Mod{\operatorname{Mod}}\def\CAlg{\operatorname{CAlg}}\def\Ab{\operatorname{Ab}}\def\Sym{\mathcal{S}}\def\V{\mathbb{V}}\def\A{\mathbb{A}}$ Let $M$ be a $k$-module and $G=\Spec H$ be an affine group scheme. After being initially...

In standard projective (P^2), the line at infinity is defined as ([X,Y,Z]) with (Z=0). However, other authors define it as ([X,Y,Z]) with (X=0). I feel some confusion. Can it be shown that this does not depend on these choices? Is the elliptic curve group structure preserved under projective...

I am tasked to prove the formal affine line $\hat{\mathbb{G}}_a$, seen as the functor from animated rings to set $$Ani(Ring)\to Ani$$ $$R\mapsto Nil(\pi_0(R))$$ taking an animated ring to the nilradical of its underlying static ring, to be an etale stack i.e. I have to show that it satisfies...

I want to read about inverse Galois Theory with the goal of proving the Hilbert Irreducibility Theorem. I do know the basics of Algebra (Group and Ring Theory, Field and Galois Theory, a bit of Moduls). Is there any good book which is on an undergraduate level about Inverse Galois Theory?

I'm trying to figure out some facts about the pullback of Cartier divisors, but honestly I'm having a hard time describing the pullback of the sheaf associated to divisor. Let $\phi\colon X \to Y$ be a dominant morphism of schemes. Suppose $X, Y$ are both integral, noetherian, separated. If...

It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of $K\subseteq L$. Therefore, $G$ gets a "natural" topology that turns it into a compact group. Why...

Recently we encountered a difficult problem in computer science, but since it is very closely related to mathematics, I was unsure which board would be more appropriate. In the end I posted it here on the mathematics board. To make the problem easier to understand, I will give both a...

Is the following statement true or false?Let $\mathbb{N}$ be the set of positive integers. For any $z > 2$, there always exist $x, y < z$ such that:$$f(x) + f(y) = f(z)$$Where the arithmetic function $f(n)$ is defined as:$$f(n) = \prod_{p^k \parallel n} \left( \frac{p^{k+1}-1}{p-1} \right) =...

As I mentioned in my previous post, I have been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. While editing that post, a second question came up, which I thought was sufficiently independent to deserve its own post. So here is the...

The discovery by Andrew Booker of an integer solution to, $$N=x^3 + y^3 +z^3=33$$ $$8866128975287528^3 - 8778405442862239^3 -2736111468807040^3=33$$ got some press and Youtube mileage back in 2019. As mentioned in Booker's July 2019 article, for $0<N<1000$, there used to be $13$ unsolved $N$,...

I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław Sierpiński's 250 Problems in Elementary Number Theory, I came across a remark stating that A. Mąkowski...

I ran a simulation where for each 2-digit combination with 30 symbols (so 0 to T), it checked, from base 2 to base 10,000, in how many bases that specific symbol combination resulted in a prime number. The top 10 were 65,6B,6H,6T,6N,61,6D,67,6J, and 6P. All starting with 6. Anyone have any idea...

This problem is from the book Álgebra comutativa em quatro movimentos by Borges and Tengan. Let $k$ be a field of characteristic not equal to $2$ or $3$. Let $f(x, y) = y^2 - x^3 - 10$ and consider the ring $R = k[x, y]/(f(x, y))$. Show that $R$ is a Dedekind domain. Edit. We also have to assume...

I am reading the paper Oscillatory Integrals with Uniformity in Parameters by Kiral, Petrow, and Young, and I am having trouble understanding the notion of an inert function introduced in Definition $2.1$. These are my confusions Since $X=X_T \in [1,\infty]$.Then if we consider a family of...

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly quotients), but I will restrict to two. Compute the Krull dimension of the following rings. $R =...

$\newcommand{\e}[1]{\exp\left(#1\right)}$ $\newcommand{\le}{\leqslant}$ In Terry Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function), Exercise 5 outlines the derivation of the Vinogradov-Korobov bound for Dirichlet $L$-functions. Let $\chi$ be a non-principal character of...

Let P(n) and Q(n) be two non-trivial polynomials in n with rational coefficients and z[P, Q] is the value of infinite sum of P(n)/Q(n) from n=1 to +∞ (only when it converges, in which the degree of Q should be larger than or equal to the degree of P plus 2). Claim: It is impossible for z[P,Q] to...