# Tagged Questions

**1**

vote

**1**answer

129 views

### Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...

**2**

votes

**1**answer

96 views

### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

**1**

vote

**0**answers

48 views

### Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials".
Let $\Lambda$ be the ring of symmetric functions over ...

**5**

votes

**2**answers

414 views

### Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...

**5**

votes

**1**answer

213 views

### A “known” Pythagorean identity in algebra?

Some will recognize this as similar to a question I asked before, but
I want to ask it without the trigonometry.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in
$x_1,x_2,x_3,\ldots$. ...

**8**

votes

**2**answers

653 views

### What is known about zero-sets of Schur polynomials?

Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one).
Let $U_\lambda^{(r)}$ be the ...

**9**

votes

**4**answers

2k views

### Expressing power sum symmetric polynomials in terms of lower degree power sums

Is there an explicit formula expressing the power sum symmetric polynomials
$$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$
of degree $k$ in $N < k$ variables entirely ...

**8**

votes

**1**answer

637 views

### When the splitting fields of shifted generic polynomials are linearly disjoint?

Let me start by rigorously pose my question.
Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + \cdots + T_n$, be the ...

**5**

votes

**1**answer

627 views

### Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function.
There exists a nice product formula for the principal specializations:
...