Timeline for English translation of “A multidimensional generalization of the Wronskian”
Current License: CC BY-SA 4.0
4 events
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| Nov 30 at 13:49 | comment | added | Joe Silverman | @D.R. The index is a valuation, so the factorization $W(f)=gh$ means that $I(W(f))=I(g)I(h)$. Then, since $g$ and $h$ involve fewer variables than $f$, one can use the induction hypothesis (and a lot of very carefully set up calculations, which is the hard part of what Roth did). Also, the entries of $W(f)$ are partial derivatives of $f$, so the indices of those entries are related to the index of $f$. Then taking the determinant, using the fact that index is a valuation, one can relate $I(W(f))$ to $I(f)$. | |
| Nov 30 at 8:55 | comment | added | D.R. | Could you share any intuition as to why the index (at a good rational approximation vector) of the generalized Wronskian $W$ (which can be factored in this way) should tell us useful things about the index of the polynomial $f$ (at that same vector)? I can follow the proof line by line, but don't have any intuition how upper/lower bounding the index of $W$ magically hands us upper bounds of $\text{ind}(f)$. For reference, I'm following gaurish4math.wordpress.com/wp-content/uploads/2015/12/… (which follows Schmidt closely). | |
| Dec 18, 2022 at 19:50 | history | edited | Joe Silverman | CC BY-SA 4.0 |
Added more explanation for how generalized Wronskians are used in Roth's theorem
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| Dec 17, 2022 at 17:42 | history | answered | Joe Silverman | CC BY-SA 4.0 |