# Tagged Questions

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### Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
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### Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3$ defined as follows: $$(z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3)$$ This map defines a Variety ($V$) of dimension $2$ in ...
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### Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3$: $$(z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3)$$ Is this map a bijection? If so, how?
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### Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
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### How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
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### Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
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### Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
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### Preimage of a point by a power map in quaternions

Suppose we have a point $x_0\in{\bf H}$ (where by $\bf H$ I denote the ring of quaternions). What I'm curious about is what can the set of solutions of $x^2=x_0$ look like? From what I've checked, ...
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### Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}.$$ If we denote by ...
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### This might be interesting. The ring $R[x_1, \dots]$ (Non-specific)

Let $R$ be a commutative ring and define $\mathcal{R} = R \oplus \bigoplus_{i=1}^{\infty} x_i R[x_1, \dots, x_i]$. Then $\mathcal{R}$ is an $R$-algebra of polynomials in any finite number of ...
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### What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
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### Questions on polynomial ring in several variables

Let $K$ be an infinite field. Prove that different polynomials in $R=K[X_1,X_2,...,X_n]$ don't lead to the same function $K^n \to K, x \mapsto f(x)$. (solved) Find $I \subset R$ and different ...
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### Is this a homogeneous polynomial

If $f(ta,tb) = f(a,b)$ $\forall t \neq 0$, then can we conclude that $f$ is a homogeneous polynomial of degree 1?
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### If $X$ is a cone, show that $I(X)$ is homogeneous.

The exercise is 1.3(3) from HP Kraft, "Appendix A: Basics from Algebraic Geometry." If a closed subset $X\subseteq \mathbb C^n$ is a cone, show that $I(X)$ is generated by homogeneous functions. ...
Let $f(x,y,z) = x^a + y^b - z^c$, where $a,b,c \gt 0$. What is the most basic yet interesting result about this polynomial from Algebraic Geometry?