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Mathematics Stack Exchange

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.

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.

253 questions
17
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1 answer

Self-study in real algebraic geometry

I would like an introductory book, a pdf or an online course to self-study real algebraic geometry. My background is the most classical one: I've already studied this book and 80% of this book. Thanks in advance EDIT1 Of course, if my background is…
user42912
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16
votes
2 answers

Global Optimization and Real Algebraic Geometry

Wikipedia suggests that: "Methods based on real algebraic geometry" are some of the "most successful general strategies" for solving global optimization problems. Could someone suggest an reference for learning about how algebraic geometry can be…
Nathaniel Bubis
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16
votes
3 answers

Two-variable polynomials over $\mathbb{Q}$ with finitely many roots in $\mathbb{R}^2$

I found this interesting exercise (not homework assignment) in real algebraic geometry: Describe an algorithm which decides whether a given polynomial in $\mathbb{Q}[X, Y]$ has infinitely or finitely many roots in $\mathbb{R}^2$. There's not much…
Bib-lost
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14
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0 answers

On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS. Definition 2. If an $n$-variate polynomial $f$ is nonnegative on $\Bbb R^n$, then $f$ is…
khashayar
  • 2,413
14
votes
0 answers

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some of my efforts: a Bernoulli lemniscate is a…
Mostafa - Free Palestine
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11
votes
4 answers

How weird can the level set of a polynomial be?

Let $p_n(\cdot)$ be a $d-$variate polynomial of degree $n$. How many connected components does the set $$\{x:\ p_n(x)>0\}$$ have? Are these components also convex? Note that, in case $d=1$ there are at most $n$ connected components, all convex, as…
Davide Maran
  • 1,259
10
votes
3 answers

Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$ $$ P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij}, $$ I would like to find a matrix $P$ which satisfies…
Alex Shtoff
  • 168
9
votes
1 answer

A function with positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)>0, \ \forall x\in \mathbb R$, then $f$ has at most $n$ roots. Context: The fundamental theorem of algebra states (when only the real…
Pavel Kocourek
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8
votes
1 answer

Real number known not to be a period

I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial motivation, namely periods. A period in my…
Florian Felix
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8
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0 answers

Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if there exist $g_{1}, \dots, g_{k} \in…
Tio Miserias
  • 160
8
votes
0 answers

When do polynomial equations come from complexification?

If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to…
Taylor
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8
votes
1 answer

Perturbing a polynomial with repeated real roots to get distinct real roots

Consider a real polynomial $f$ of degree $d$ which has $d$ real roots not necessarily distinct. In general, can we accomplish the following? For every $\epsilon>0$, can we perturb each coefficient of $f$ by less than $\epsilon$ and guarantee real,…
Suana
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8
votes
1 answer

Are algorithms for elimination of quantifiers over the reals practical?

I wanted to find the semialgebraic set in the $(a_0,a_1,a_2,a_3)$ space that guarantees that there exists at least one real root of the general polynomial equation of degree 4. For that purpose, installing QEPCAD on Ubuntu 14, I tried to eliminate…
user64540
7
votes
2 answers

A positive polynomial of Schur as a sum of squares of polynomials

This is a follow-up this question on MathOverflow. The previous time I asked about SOS was Choi Lam homogeneous polynomials as sums of squares. With positive integer $k,$ and real variables $x,y,z$ that are allowed to take on positive or negative…
Will Jagy
  • 146,052
7
votes
1 answer

Number of connected components of zero set of polynomial with bounded number of terms

Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is a polynomial of degree $\ell$. Then, the number of connected components of its zero set $\{a\in\mathbb{R}^d : f(a) = 0\}$ is bounded by roughly $\ell^{d}$. I've seen this result attributed to Warren,…
tc1729
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