Timeline for Resultant of system with 3 polynomials and 3 variables
Current License: CC BY-SA 3.0
13 events
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| Sep 20, 2018 at 11:19 | comment | added | Joe Silverman | @Freeman. There is no algebraic expression of the coefficients whose vanishng implies a real solution. I guess you're asking if there are inequalities, for example for the polynomial $ax^2+bx+c$, there is an inequality involving the discriminant which says if there is a real root. I do not know the answer in general. But if you look at a textbook on real algebraic geometry, this question is likely to be discussed. | |
| Sep 20, 2018 at 5:26 | comment | added | Turbo | @JoeSilverman Thank you that helps. So is there a calculation of resultant that works in $\mathbb P^2(\mathbb R)$? | |
| Sep 20, 2018 at 0:54 | comment | added | Joe Silverman | @Freeman. Yes, that's why I said a non-trivial zero in projective 2-space, which rules out the trivial zero at $(0,0,0)$. It also identifies zeros that are scalar multiples of one another, which is the natural way to count zeros in this setting. | |
| Sep 19, 2018 at 23:06 | comment | added | Turbo | @JoeSilverman $f_1,f_2,f_3$ are homogeneous and so they have a trivial zero at $(0,0,0)$. Resultant is zero iff $f_1,f_2,f_3$ vanish at some $p\not=(0,0,0)$? | |
| Sep 29, 2014 at 23:33 | comment | added | Jason DeVito - on hiatus | @JoeSilverman: I see. Thank you for all your help! | |
| Sep 29, 2014 at 23:22 | comment | added | Joe Silverman | @JasonDeVito I think you have to homogenize, then check if the common solutions are points "at infinity". For example, do $ax+b$ and $cx+d$ have a common root. The only sensible polynomial condition to check is $ad-bc=0$. But this says that they have a common root if $a=c=0$, which is true only in the sense that if $a=c=0$, then they both vanish at the point at infinity in $\mathbb P^1$. | |
| Sep 29, 2014 at 20:01 | comment | added | Jason DeVito - on hiatus | (Rereading your original post, I now see the first sentence in the second paragraph is only referring to homogeneous systems, not all polynomial systems. Sorry for misreading!) | |
| Sep 29, 2014 at 20:00 | comment | added | Jason DeVito - on hiatus | I'm sorry to be a bother, but those references seem to only handle the case of homogeneous polynomials. Do you know of a reference for the inhomogeneous case? Precisely: Given $f_1,..., f_m\ in k[x_1,...,x_n]$ with $k$ algebraically closed and $m > n$, is there a polynomial ideal in the coefficients of the $f_i$ which vanishes iff the polynomials have a common root in $k^n$? (My understanding is that any system of polynomials can be homogenized, and then one can use the Macaulay resultant. But this resultant may vanish due to projective solutions even when there is no solution in $k^n$). | |
| Sep 29, 2014 at 17:10 | comment | added | Jason DeVito - on hiatus | @Joe: Thank you! I'm a differential geometer by trade, but find my research bumping up against algebraic geometry. I'll be sure to check out all those references! | |
| Sep 29, 2014 at 17:05 | comment | added | Joe Silverman | @JasonDeVito Pretty much any book that has a section on "Elimination Theory" will contain this result. I first saw it in van der Waeden's "Algebra", which is a bit old-fashioned, but beautifully written. Or see Eisenbud's Commutative Algebra with a view towards Algebraic Geometry, Chapter 14, Theorem 14.1. The proof is in the exercises starting on page 318, and he refers to Mumford's Complex Projective Varieties for those who want to look up the proof. | |
| Sep 29, 2014 at 15:52 | comment | added | Jason DeVito - on hiatus | Do you have a reference for the first statement in your second paragraph? I'm particularly interested in the case where we have $n+1$ inhomogeneous polynomials in $n$ variables, over $\mathbb{C}$. (Or perhaps I'm misunderstanding the scope of that sentence.) | |
| Feb 4, 2014 at 13:33 | comment | added | Abdelmalek Abdesselam | yet it is a principal ideal | |
| Feb 3, 2014 at 22:41 | history | answered | Joe Silverman | CC BY-SA 3.0 |