# Tag Info

• 141k
Accepted

• 2,175

Accepted

### Singular locus of zero set of elementary symmetric polynomial

Notice that the vanishing of $\frac{\partial}{\partial x_1} \sigma_{m,r}, \cdots, \frac{\partial}{\partial x_m} \sigma_{m,r}$ implies the vanishing of $\sigma_{m,r}$ since \sigma_{m,r}=\frac{1}{r}\...
• 85.2k
Accepted

### Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?

The notation and framing of the problem are not optimal, because the $n_i$'s being integers is a bit of a distraction. Let $[k]:=\{1,\ldots,k\}$, and let ${\rm Part}_k$ be the set of set partitions of ...

### Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The answer by Jules Lamers considers $k=1$. In the comments to that answer, people have mentioned that the case $k=0$ has been solved. Namely, for a given partition $\lambda$ let Del($\lambda$) be the ...
• 1,519

### When is a linear combination of the elementary symmetric polynomials reducible?

Suppose that $f=p_1 p_2\cdots p_k$, where each $p_i$ is irreducible. Since $f$ is a symmetric function, for every $w\in S_n$ and every $i$ we must have $w\cdot p_i = cp_j$ for some $j$, where $c$ is a ...
• 49.9k

### Inequality with symmetric polynomials

This is easily proven using the Rearrangement Inequality, which says that if we have two sequences of reals, and we are to pair them up in such a way as to maximize the sum of the products of the ...
• 4,745
Follows from HÃ¶lder's inequality (p=6, q = 6/5): $ab^5 + ba^5 \le (a^6+b^6)^{1/6} (b^6+a^6)^{5/6}$
Under the specialization $\mathcal{S} = \{1, \frac12, \frac13, \ldots, \frac1n \}$, the corresponding products to not converge to a limit as $n\to\infty$ since the harmonic series diverges. However, ...