What is known about zero-sets of Schur polynomials? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:41:34Z http://mathoverflow.net/feeds/question/76859 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76859/what-is-known-about-zero-sets-of-schur-polynomials What is known about zero-sets of Schur polynomials? Symm 2011-09-30T14:53:24Z 2012-03-21T22:22:00Z <p>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). </p> <p>Let $U_\lambda^{(r)}$ be the zero-set in $\mathbb{C}^r$ of the Schur polynomial $s_\lambda(x_1,\cdots,x_r)$. </p> <p>What is known about $\cap_{\lambda \in S} U_\lambda^{(r)}$, beyond the fact that it is symmetric under the action of $S_r$?</p> <p>(I am having trouble finding information about this: all the hits are about the different question of the Schur stability of univariate polynomials, a concept based on the location of the roots of those polynomials).</p> http://mathoverflow.net/questions/76859/what-is-known-about-zero-sets-of-schur-polynomials/87841#87841 Answer by Per Alexandersson for What is known about zero-sets of Schur polynomials? Per Alexandersson 2012-02-07T21:24:56Z 2012-02-08T18:39:08Z <p>Assume there is a non-trivial common zero $\xi \neq (0,\dots,0)$.</p> <p>Now, to quite wikipedia: "The degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables."</p> <p>So, let $P$ be any symmetric homogeneous polynomial in $n$ variables, of degree $d$. It is now clear that $P$ is a linear combination of the Schur polynomials, hence, $P$ must be zero at $\xi.$</p> <p>Edit: This seems very strange, at least when the number of equations is greater than the number of variables. For small $r$, there might be a few zeros except 0, but when the number of partitions of $r$ is greater than $r$ itself, then one cannot apriori expect a solution except 0.</p> http://mathoverflow.net/questions/76859/what-is-known-about-zero-sets-of-schur-polynomials/89226#89226 Answer by John Jiang for What is known about zero-sets of Schur polynomials? John Jiang 2012-02-22T20:37:02Z 2012-02-22T20:37:02Z <p>For $k :=|\lambda| \ge r$, the statement that all $s_\lambda(x_1, \ldots, x_r)$ vanish is equivalent to all the elementary polynomials $e_j(x_1, \ldots, x_r) := \sum_{i_1 &lt; \ldots &lt; i_j} x_{i_1} \ldots x_{i_j}$, $j \le r$, vanish, since the latter form a basis of the algebra $\Lambda_r$. But this is true if and only if all the $x_1, \ldots, x_r$ are zero, since $e_j$ is the $y^{r-j}$ coefficient of a degree $r$ polynomial, i.e., $$\prod_{i=1}^r (y - x_i) = \sum_{j=0}^r (-1)^j e_j(x_1, \ldots, x_r) y^{r-j}.$$ If all the $e_j$'s are zero, except $e_0 \equiv 1$, then it must be $y^r$, hence all the $x_i$'s vanish.</p> <p>For $k &lt; r$, the zero set consists of roots of polynomials of the form $y^r + c_{k+1} y^{r-k-1} + c_{k+2} y^{r-k-2} + \ldots + c_r$.</p>