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1
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1answer
29 views

About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
3
votes
0answers
85 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
1
vote
1answer
65 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
2
votes
0answers
81 views

Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...
1
vote
1answer
48 views

image under a morphism of a variety defined over R

If you have say an affine variety defined over $\mathbb{R}$, then its image under a morphism (also defined over $\mathbb{R}$) is a constructible set. But presumably there would be no good reason in ...
0
votes
0answers
46 views

homogeneous polynomials of degree 3 in two variables

I am interested in trying to determine if the following statement is true. Let $V$ be the four-dimensional real vector space consisting of all homogeneous polynomials of degree 3 in two variables ...
2
votes
1answer
115 views

Determine whether a system of polynomials with real coefficients has a real solution?

Today my students asked me the following problem: Define polynomials $P_j$ with real coefficients $$P_{j}=\sum_{i_{1},\cdots,i_{s}}a^{(j)}_{i_{1}\cdots i_{s}}X^{i_{1}}_{1}\cdots X^{i_{s}}_{s}, ...
2
votes
0answers
59 views

conic structure at infinity for non-closed unbounded semi-algebraic sets

Let $X\subseteq\mathbb{R}^k$ be a non-closed, unbounded semi-algebraic subset. Then it seems to me that Proposition 5.49 on p. 189 of the book Algorithms in Real Algebraic Geometry still holds true ...
7
votes
0answers
121 views

Nearest Point to a real algebraic set

Suppose I have a compact bounded real algebraic (eventually: or analytic or semialgebraic or semianalytic set) $V$ in $\mathbb R^3$ and a point $x\in\mathbb R^3$ not in $V$. How much do we know about ...
-1
votes
1answer
140 views

Smooth algebraic functions

Suppose $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are ...
0
votes
0answers
90 views

Dimensions of two spaces

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$. Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ ...
2
votes
1answer
165 views

How many circular cylinders through 5 general points?

How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
2
votes
3answers
205 views

Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant ...
6
votes
0answers
209 views

Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows: Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...
4
votes
1answer
139 views

The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
1
vote
0answers
64 views

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 ...
2
votes
1answer
120 views

Non-negativity of invariant polynomials

Certifying the non-negativity of a symmetric polynomial is much easier than certifying the non-negativity of an arbitrary polynomial function: for instance, in (1) is proved that the complexity of ...
0
votes
0answers
36 views

Real centrally-symmetric plane algebraic curves

I am looking for a reference regarding the topology of real centrally-symmetric plane algebraic curves. By this I mean the curves defined by $$ P(x,y)=0, $$ where $P$ is a degree $m$ polynomial, ...
2
votes
0answers
70 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
5
votes
1answer
218 views

Connected components of the complement of a degree-d affine hypersurface

Let $n$ and $d$ be positive integers, and $f\in\mathbb{R}[x_1,\dots,x_n]$ be a polynomial of degree $d$. Let's consider the zero-set $M = \{x \in \mathbb{R}^n: f(x) = 0\}$ of $f$. Can we estimate ...
3
votes
1answer
117 views

Handelman's positivstellensatz for symmetric matrix-valued polynomials

For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$. Several ...
3
votes
1answer
98 views

Tropical polynomial Positivstellensatz

In real algebraic geometry, Stengle's Positivstellensatz can be used to characterize polynomials that are positive on a semialgebraic set. Say that a tropical semialgebraic set is a subset of ...
7
votes
3answers
247 views

nonnegativity conditions for a polynomial in two variables

Let $$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$ be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$. What are the ...
5
votes
1answer
216 views

Locally Closed Orbits in Real Algebraic Geometry

Let $G$ be a real algebraic group, and let $X$ be a real affine $G$-variety. I am looking for conditions on $G$ and $X$ for which the $G$-orbits are known to be locally closed in the Zariski topology ...
3
votes
1answer
112 views

Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows: Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$ such that ...
8
votes
2answers
131 views

Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ : $$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$ The discriminant of this polynomial in $r$ is the following expression (obtained using ...
6
votes
1answer
200 views

cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...
4
votes
2answers
235 views

Approximation of curves

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
3
votes
1answer
125 views

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 ...
6
votes
0answers
126 views

Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
6
votes
0answers
59 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
10
votes
0answers
179 views

real algebraic geometry software?

Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
7
votes
3answers
439 views

Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$). We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...
4
votes
0answers
94 views

moduli in real/semi algebraic geometry

Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry? The sort of thing I am imagining is an object in a category of semischemes: Ordinary schemes ...
10
votes
1answer
645 views

What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$. Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
1
vote
0answers
85 views

Representing quasianalytic functions in several variables

For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
40
votes
7answers
5k views

How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...
2
votes
1answer
90 views

Compactness of a semi algebraic set

Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?
5
votes
1answer
187 views

Simple criterion to verify that the real zeros are an irreducible algebraic set

Given an irreducible polynomial $p$, its set of real zeros might be a reducible algebraic set. For example: $p=(x^2-1)^2 +(y^2-1)^2$. Is there a simple sufficient condition on $p$ so that its real ...
1
vote
1answer
148 views

Homogenous polynomial inequalities to prove positivity?

If I have $n$ variables $x_1, \cdots, x_n$, and a set $S$ of inequalities of the form $p(x_1, \cdots, x_n) > 0$ where $p$ is a homogenous real polynomial, is it true that I need $|S| \geq n$ to ...
1
vote
0answers
42 views

Components of Intersection of Ellipsoids

Let $\Sigma$ be an $n-1$ dimensional ellipsoid in $\textbf{R}^{n}$ and $S$ the unit sphere. I would like to understand the connected components $C$ of the intersection of $\Sigma$ and $S$. In my ...
5
votes
1answer
197 views

Are there general position results in singular algebraic sets?

Let $X$ be a real algebraic set, and let $Y \subset X$ be its singular set. In this question I'll focus on the analytic topology, so we can just imagine that $X$ is the zero set, in $\mathbb{R}^n$, ...
3
votes
1answer
217 views

The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...
2
votes
1answer
163 views

Volumes of families of semialgebraic sets

Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is ...
2
votes
3answers
179 views

How to tell if a second-order curve goes below the $x$ axis?

Suppose we have a second-order curve in general form: (1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$. I'd like to know if there is a simple condition that ensures that the curve ...
6
votes
0answers
219 views

Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case. I call a partition ...
0
votes
1answer
134 views

Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates): Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of ...
6
votes
2answers
356 views

orderings of the field R((x, y))

I don't know much about the theory of ordered fields. But I know that, for the real fields $\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$, we can explicitly determine all the ...
4
votes
1answer
438 views

Exact arithmetic for real algebraic numbers

There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...
1
vote
1answer
83 views

Decidability of the generated order

is the question whether a polynomial is non-negative on some semi-algebraic set (equivalently, is it in the cone denerated by some polynomials in the field of rational functions) known to be ...