It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with $\sigma_{k}(x) = 0 = \sigma_{k+1}(x)$, where $\sigma_{k}(x)$ denotes the $k$-th elementary symmetric function of the coordinates of $x$, is given by exactly the points with $n-k+1$ coordinates equal to zero. In some sense, this says that $\Lambda_{k}$ is "trivial" in the sense that $\sigma_{k}$ and $\sigma_{k+1}$ are both zero if and only every single term in both $\sigma_{k}$ and $\sigma_{k+1}$ vanish (i.e. each term includes a zero coordinate).
I am trying to determine if this is "approximately true"--that is, if $|\sigma_{k}|, |\sigma_{k+1}|$ are both sufficiently small, are we reasonably close to $\Lambda_{k}$? How close? More precisely,
Can we (explicitly) find a function $f(\epsilon_{1},\epsilon_{2})$ so that for any point $x \in \mathbb{S}^n \subset \mathbb{R}^{n}$ with $|\sigma_{k}(x)| < \epsilon_{1}$, $|\sigma_{k+1}(x)| < \epsilon_{2}$, some $n-k+1$ coordinates of $x$ are bounded in absolute value by $f(\epsilon_{1},\epsilon_{2})$?
Note that I restrict to $\mathbb{S}^{n}$ to prevent any effects due to scaling/homogeneity of the $\sigma_{k}$.
I am trying to show that under the hypotheses stated immediately above, if $\epsilon_{1} < \beta^{2+\alpha}, \epsilon_{2}<\beta^{1+\alpha}$ with $\beta$ small, then we have $\sigma_{k+l} = o(\beta^{l-2+\alpha})$ as $\beta\to 0$ for $l \ge 2$. Such a result would follow if a sufficiently small $f$ could be found for the question above. Based on numerical tests I have done (and the result I stated at the beginning) it seems very likely that such an $f$ can be found.
One approach would be to simply try to determine how quickly the zero sets of $\sigma_{k}$ and $\sigma_{k+1}$ separate away from $\Lambda_{k}$. I've tried looking at the series expansions of $\sigma_{k},\sigma_{k+1}$ near $\Lambda_{k}$ to study such separation but have had difficulty obtaining useful information this way (the expressions I obtain become complicated for $k$ at all large).
I suspect there may be some general result about the how quickly zero sets separate away from their intersection (especially for polynomials of different degrees), but I am not aware of such a result (I work in differential geometry). Thank you for any thoughts you may have.
