Linked Questions

29
votes
2answers
1k views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
16
votes
3answers
425 views

Matrix function converges, how about the eigenvalues?

Suppose I have a matrix function $A(t)$ with $$\lVert A(t) - B\rVert \le ct^\alpha$$ in some matrix norm (this will work for any norm, I guess). So, in a sense $A(t)\rightarrow B$ for $t\rightarrow 0$ ...
9
votes
4answers
897 views

On continuity of roots of a polynomial depending on a real parameter

Problem Suppose $f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]$ for all $t\in\mathbb R$, where $a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C$ are continuous on $t$...
8
votes
2answers
309 views

The map that sends $A$ to its greatest eigenvalue is continuous.

The map $f:S_n(\mathbb R)\to \mathbb R$ such that $f(M)$ is the greatest eigenvalue of $M$ is continuous ($S_n(\mathbb R)$ is the set of symmetric matrices) I need to prove this result in order to ...
5
votes
2answers
1k views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...
7
votes
3answers
388 views

The Riemannian Distance function does not change if we use smooth paths?

The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all piecewise smooth paths between $p$ and $q$. Does it change if we take the infimum only over smooth paths?...
3
votes
1answer
677 views

The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial? [closed]

Why the following is true? «The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial?»
3
votes
1answer
990 views

Disjoint Gershgorin disks $\Rightarrow$ each contains exactly one eigenvalue

It is an exercise in Peter Lax's book Linear Algebra that if all the Gershgorin disks $$D_i := \{z\in \mathbb{C} : |a_{ii} - z| \leq \sum_{i \neq j} |a_{ij}|\}$$ are disjoint, then each disk must ...
5
votes
2answers
228 views

Quotient space of $\Bbb C^n$ obtained by action of $S_n$

Consider the action of $S_n$ on $\mathbb{C^n}$ given by: $$\sigma(x_1, x_2, \cdots,x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \cdots,x_{\sigma(n)}).$$ What is the quotient space of $\mathbb{C^n}$ obtained ...
8
votes
1answer
252 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
9
votes
0answers
269 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
1
vote
1answer
185 views

The roots of a polynomial are a continuous function of the coefficients

What does the following mean? The roots of a polynomial are a continuous function of the coefficients Please guide me with an example.
3
votes
1answer
37 views

Is the application which associates a polynomial with its root continuous?

Let $f:\mathbb{R}^{2n+1}\to\mathbb{R}$ be defined through: $f\left(x_0,...,x_{2n}\right)$ is the greatest root of the polynomial $p(t)=\sum_{k=0}^{2n}x_kt^k$. Is $f$ continuous? If so, what is the ...
2
votes
1answer
81 views

Question on continuity of elementary symmetric polynomials

I am not being able to find a clear answer to the following question with my limited knowledge of algebra. Let us consider $n$ real-valued functions {$f_{i}(t): t \in \mathbb{R}$ and $ i={1,2,3,...,...
3
votes
1answer
122 views

When are the limits of roots of a polynomial identical to the roots of the limit of the polynomial?

I have a univariate polynomial of degree $n$ (where $n$ is larger than $4$). The real-valued coefficients of the polynomial depend on a parameter $\psi$, i.e. $$p_\psi(x)=a_n(\psi) x^n+a_{n-1}(\psi) x^...

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