Timeline for which homogeneous polynomials split into linear factors?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2018 at 12:27 | history | edited | Adam Przeździecki | CC BY-SA 3.0 |
one "\" too many
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| Oct 11, 2012 at 15:26 | comment | added | Will Traves | @Fernando: Can't we view the set of completely reducible homogeneous polynomials of degree n in d variables as the image of the multiplication map sending (P(S1))n to P(Sn), where S=C[x0,…,xd−1]? The image of this morphism should be a projective subvariety of P(Sn), in particular it is a Zariski-closed set. The affine cone over this subvariety gives the affine variety that Jeurgen described. | |
| Oct 11, 2012 at 2:26 | vote | accept | Mark C. Wilson | ||
| Oct 11, 2012 at 2:27 | |||||
| Oct 10, 2012 at 23:01 | comment | added | Fernando Muro | This answer may be incorrect. The equations $F_\mu=0$ describe the Zariski closure of the sets of polynomials which factor as a product of linear factors, but nothing guarantees that this set is Zariski closed. It is of course constructible for the Zariski topology, since it is the image of a map between affine schemes (this is essentially what you show in your answer), but I don't see why it should be Zariski closed. BTW, Gröbner bases are a tool to compute the elimination ideal, which is simply an intersection, but conceptually they are unnecessary. | |
| Oct 10, 2012 at 22:38 | history | answered | Jürgen Böhm | CC BY-SA 3.0 |