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Results tagged with real-algebraic-geometry
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user 36563
Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).
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image under a morphism of a variety defined over R
You are right that in general this is not the case. For example let $X$ be the zero set of $x-y^2$ in $\mathbb{A}^2$ and consider the projection $X \to \mathbb{A}^1, (x,y) \mapsto x$.
Perhaps you are …
- 3,031
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Handelman's positivstellensatz for symmetric matrix-valued polynomials
The obvious generalization is false: Consider $A=\begin{pmatrix} 1+x &y \\y & 1-x \end{pmatrix}$ which is a linear matrix polynomial. It defines a compact set in the plane, namely the unit disc. The a …
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Simple criterion to verify that the real zeros are an irreducible algebraic set
Let $p \in \mathbb{R}[x_1, \ldots, x_n]$ be irreducible. If there is a point $v \in \mathbb{R}^n$ such that $p(v)=0$ and sucht that the gradient $\nabla p(v) \neq 0$. Then, by the Artin-Lang-Theorem, …
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Hilbert scheme of real curves
Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise.
Let $H$ be the Hilbert s …
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Smooth, irreducible surface with real part containing two projective planes
Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are …
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answer
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Sum of Squares Length of a Product
Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer suc …
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vector space of ternary forms with real rooted property
Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that eve …
- 3,031
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About preserving real-rootedness of multivariable polynomials
Consider the univariate case:
In general, a linear combination of real-rooted polynomials is not real-rooted: Let $f_1=(z+1)^2$ and $f_2=(z-1)^2$. Every convex combination of $f_1,f_2$ which is not $ …
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Compactness of a semi algebraic set
Let $p \in \mathbb{R}[x_1, \ldots, x_n]$. The set $S=\{x \in \mathbb{R}^n: p(x) \geq 0\}$ is compact if and only if there is a natural number $N$ and polynomials $g_i, h_i \in \mathbb{R}[x_1, \ldots, …
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