36
votes
Accepted
$\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?
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- 128k
35
votes
Accepted
Can a real quartic polynomial in two variables have more than 4 isolated local minima?
Recent addition: Inspired by DimaPasechnik's and Matt F.'s comments about sum of squares decompositions, I tried the following very natural idea: Try to find $f$ of the form $f(x,y)=A(x,y)^2+B(x,y)^2$,...
- 22.5k
27
votes
Accepted
Positive quadratic polynomial
No. The main idea is to find a $q$ on $S$ that is the sum of squares, but where the expressions being squared are not linear combinations of the coordinates in ${\bf R}^n$.
Consider the space curve
\...
- 114k
25
votes
Accepted
Is the "equidistant curve" to an algebraic curve algebraic?
Yes, $L_\delta$ is algebraic. You can find its equations by elimination theory as follows: Let $L$ be defined by the polynomial equation $F(x,y) = 0$. Now consider the polynomial equations
$$
F(x,y)...
- 108k
21
votes
Accepted
Positive 4-form
(6 October 2023) I'll leave the original argument below because it seems that many people liked it, but, in fact, it wanders around and introduces a lot of unnecessary information, which obscures the ...
- 108k
18
votes
Accepted
Rigid non-archimedean real closed fields
Charles Steinhorn and I have answered this question positively by constructing a rigid non-archimedean real closed field of transcendence degree 2. Our preprint is now posted on arxiv.
https://arxiv....
- 3,530
18
votes
Is every minimal hypersurface in $S^n$ algebraic?
The answer to this question is 'no' for most minimal surfaces of revolution in the $3$-sphere.
Consider surfaces in $S^3 = \{\,(z,w)\in\mathbb{C}^2\,|\,|z|^2+|w|^2=1\,\}$ that are invariant under the ...
- 108k
17
votes
Accepted
Differentiability of eigenvalues of positive-definite symmetric matrices
In the open subset of $M_n(\mathbb{R})$ where the $\lambda_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem.
On the other hand, when some eigenvalue ...
- 38k
16
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
The keyword is the Cartan decomposition in the theory of symmetric spaces.
In short, when an eigenvalue is simple (its multiplicity is $1$) it is locally an analytic function. But when the ...
- 2,803
16
votes
Accepted
Maximization of a cubic form over the $14$-dimensional sphere
Put $x_{ij}=x_{ji}$ and consider the symmetriс matrix $A=(x_{ij})$ with zeros on diagonal. Then $\operatorname{tr} A=0$, $\operatorname{tr} A^2=2\sum_{i<j} x_{ij}^2=2$ is fixed, denote it $2=30 \...
- 109k
14
votes
Accepted
Can a cubic polynomial in two real variables have three saddle points?
The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-...
- 22.5k
14
votes
Accepted
How many saddle points can a quartic polynomial in two real variables have? All 9?
By (3.1) of Counting Critical Points of Real Polynomials in Two Variables by Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby a degree $d$ polynomial with only nondegenerate critical ...
- 148k
14
votes
Connеcted components of irreducible algebraic varieties
For a (projective) smooth real plane curve $C \subset \mathbb{RP}^2$ the answer is known.
Such a curve is a compact smooth one-dimensional manifold without a boundary, so its connected components are ...
- 66.3k
13
votes
Accepted
Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?
The answer is negative: In general $\psi({\rm End}(V))$ is not closed. Here is a proof when $d\ge4$ is even and $k=d-1$. Notice that $\Lambda^{d-1}V$ can be identified with $V$, so that $\psi(A)$ is ...
- 52.3k
13
votes
Accepted
Counting real zeros of a polynomial
By Remark 9.21 page 340 of the book by Basu, Pollack and Roy on real algebraic geometry the matrix $H$ is the expansion of the Bezoutiant of $P$ and $P'$ in the Horner basis of $P$ instead of the ...
13
votes
Accepted
What is the topology on the set of field orders
The topology you are looking for is called the Harrison topology. If we denote the set of ordering of a field $F$ with $\mathrm{Sper}\,F$ (more on that in a moment), this is the subspace topology ...
- 16.5k
13
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
As mentionned by other answers, simple eigenvalues are $C^\infty$, while non-simple ones are not. Let me add however two important properties which you can find in Kato's book Perturbation theory of ...
- 52.3k
13
votes
Accepted
An elementary inequality for three complex numbers
I will prove the original inequality.
First, performing the change of variables $x=1/a$, etc., and inverting the harmonic mean, we need
$$
\sum \left|\frac{yz}{x(y+z-x)}\right|\geq \frac32.
$$
Next, ...
- 23.7k
12
votes
Accepted
Every real variety contains non-singular points
If you're willing to view a variety as a set of points rather than as a scheme, then it is fairly easy to show that every real algebraic variety $V$ in ${\bf R}^n$ is equal as a set to the finite ...
- 114k
12
votes
Accepted
General Tarski-Seidenberg Theorem
The most abstract version of the Tarski-Seidenberg theorem I know of is the following
Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map
$$f^*:\...
- 16.5k
12
votes
Accepted
How does complex conjugation act on the Hodge filtration?
Let us first consider the singular cohomology group $H^i(X(\mathbb{C}),\mathbb{C})$. There are three ways to let the group $G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2$ act on it : by ...
- 6,543
12
votes
Accepted
Is integration semi-algebraic?
Consider the function $$f(x,y) =\begin{cases} x & \text{if }1\leq x\leq 2 \text{ and }0\leq y\leq \frac{1}{x}\\
|x|+|y|+2& \text{otherwise}\end{cases}$$
Then $f$ is semi-algebraic, $\{f\leq t\}...
- 5,031
11
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
Let us consider functions $A$ from (an open interval in) $\mathbb{R}$ into the set of symmetric real $n\times n$ matrices (Hermitian complex $n\times n$ matrices behave analogously).
If $A$ is given ...
- 648
11
votes
Accepted
An almost complex structure on the real $n$-sphere $S^n$
Let $M=SU_3$, the compact semisimple Lie group.
By request of the OP, for those unfamiliar with Maurer-Cartan form, let me define it. Write each point of $SU_3$ as a matrix $g$. Left translation by $g^...
- 26.3k
10
votes
How can I distinguish a genuine solution of polynomial equations from a numerical near miss?
Interval/ball arithmetic may help, actually.
It can be used to prove existence of solutions to multivariate systems like this one. The main idea is: reformulate your system as a fixed-point system $x ...
- 20.2k
10
votes
Accepted
Cohomology of real analytic coherent sheaves
In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's Analytic cycles on complex manifolds.
For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri ...
- 1,274
10
votes
Elementary inhomogeneous inequality for three non-negative reals
Write $x = 1-X$, $y=1-Y$, $z=1-Z$. Then the inequality reduces to
$$2XYZ \leq X^2 + Y^2 + Z^2$$
for $X, Y, Z \leq 1$. If $X, Y, Z < 0$ then the inequality is trivial, since LHS < 0. Otherwise ...
- 9,617
10
votes
Accepted
Nowhere negative polynomials form a semialgebraic set
As I said in the comments this is very well known: $S$ is the complement of the projection of the semialgebraic set $\{(f,a)\in P_{d,n}\times\mathbb{R}^n:f(a)<0\}$, hence semialgebraic by the ...
- 2,051
10
votes
Accepted
Projections of compact real algebraic sets
The answer is no. (Although the previous example I gave was bad.)
Let $C$ be the curve $y^2 = x^2 (x-1)(2-x)$, so $C$ has a smooth component with $1 \leq x \leq 2$, and also a node at $(0,0)$. Let $M$ ...
- 156k
10
votes
Accepted
Degree four polynomials with no real roots
The answer is contained in this wonderful paper:
E. L. Rees. Graphical Discussion of the Roots of a Quartic Equation. The American Mathematical Monthly, Vol. 29, No. 2 (Feb., 1922), pp. 51-55
In which ...
- 326
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