# Questions tagged [real-algebraic-geometry]

Real algebraic geometry is the study of algebraic geometry over the real numbers, or more generally formally real (esp. real closed) fields. Problems in this tag may require a mix of methods from algebraic geometry and techniques from o-minimal (esp. semialgebraic) geometry.

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### When do two plane cubic curves have 9 real intersection?

What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an ...
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### Concerning the ring of continuous functions on $\mathbb{R}$

It is not difficult to check that the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a ring (an $\mathbb{R}$-algebra), and similarly (if I am not wrong), the set of continuous ...
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### Simultaneous real solution of $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$

I am trying to solve the following system of non-linear equations in real numbers: $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$, with $x,y$ real. I can only see that $xy\ne 0$. I have no clue whether a ...
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### Is the minimizer of the distance from a point to a closed set generically unique?

Let $\mathcal{C} \subset \mathbb{R}^n$ be a closed set and let $E$ be the set of points in $\mathbb{R}^n$ for which there is not a unique closest element of $\mathcal{C}$. That is, if $x \in E$, then ...