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Suppose $V$ is an affine algebraic set defined by real polynomials. Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with Zariski topology where the closed sets are algebraic sets (in $\math …

I have learned that if $G$ is an algebraic group and $H$ is a normal closed subgroup then $G/H$ is also an algebraic group satisfying: for any morphisms $\phi : G \rightarrow X$ constant on the classe …

Let $f: X \rightarrow Y$ be a morphism of schemes: $X$, $Y$ are regular schemes, let $Z_1, Z_2$ be two closed regular subschemes of $Y$, let $x \in X \times_Y Z_2$ such that $y = f(x) \in Z_1 \cap Z_2 …

Let $F$ be a homogeneous from in $\mathbb{R}[x_0, .., x_n]$. Then $F$ defines a projective variety $X \subset \mathbb{P}_{\mathbb{C}}^n$. Assume $X$ is smooth. In this case $F=0$ also defines a subman …

Let $F_i$ be homogeneous forms with $\mathbb{C}$ coefficients in $n$ variables for each $i$. Let $$ T_2 = \{ (\mathbf{x}, \mathbf{y}) \in \mathbb{A}_{\mathbb{C}}^{2n} : F_i(x_1y_1, ..., x_n y_n) =0, …

Let $F_1(\mathbf{x}, \mathbf{y}), \ldots, F_r(\mathbf{x}, \mathbf{y})$ be bihomogeneous polynomials with rational coefficients with bidegree $(d_1, d_2)$, which means $$ F_i( s x_1, \ldots, s x_{n_1} …

Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(y_1, …

Suppose I have an irreducible affine variety $X \subseteq \mathbb{A}^n_k$. Let us denote $X = \{ x \in k^n : f_j(x) = 0 \ (1 \le j \le M) \}$. $k$ is an algebraically closed field. Let $a_i \in k$, $ …

Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions i …

Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers. Consider the system of congruences $$ G_j(\mathbf{x}) \eq …

Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that l …

In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows. Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that …

I'm reading this set of notes and I'm trying to understand this passage where they explain how to put a topology on $GL_n(R)$ when $R$ is a topological ring, which I am not completely following. The f …

I am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\mathbb{R}^n$. Let …

Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials. Suppose $V \subseteq \mathbb …