Timeline for Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?
Current License: CC BY-SA 3.0
10 events
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| Jun 4, 2017 at 15:47 | comment | added | user237522 | @Will Sawin, thank you! (if one of MO users can "translate" the algebraic geometry answer to a pure algebraic answer, that would be great! although I know that it is important to try to understand the algebraic geometry answer). | |
| Jun 4, 2017 at 15:13 | comment | added | Will Sawin | @user237522 For a specific family of $w$s, the first step would be to understand the genus of the geometric generic fiber and try to calculate its automorphism group. The next would be to determine whether the automorphisms extend to the whole space. | |
| Jun 4, 2017 at 13:59 | comment | added | user237522 | BTW, perhaps the reducible $w$ is not so arbitrary, because (in what I had in mind) there exists (an irreducible) $u$ which is integral over $\mathbb{C}[w]$. | |
| Jun 4, 2017 at 13:55 | comment | added | user237522 | Thanks for your explanation. Please, do you see any hope to find such $f$ for a specific family of $w$'s? | |
| Jun 4, 2017 at 12:26 | comment | added | Will Sawin | @user237522 A quartic polynomial where all monomials of degree at most $4$ appear, and all are independent transcendentals, does the trick. However a polynomial with random integer coefficients in some large box also does the trick with high enough probability. Most likely something reducible works - I doubt there are any nontrivial automorphisms of $(y^2- x^3 - x - 7) (3y-5x + 12)$, for instance. | |
| Jun 4, 2017 at 10:05 | comment | added | user237522 | Is it possible to describe all polynomials $w$ for which there do exist $f$ as I wish? Actually, I prefer to further assume that $w$ is reducible, like $x^4+y^4=(x^2−iy^2)(x^2+iy^2)=\ldots$. | |
| Jun 4, 2017 at 9:52 | comment | added | user237522 | @Jason Starr, please, what is a general quartic polynomial? | |
| Jun 4, 2017 at 9:38 | comment | added | Jason Starr | @user237522. Your degree $4$ polynomial is not a "general" quartic polynomial. | |
| Jun 4, 2017 at 6:26 | comment | added | user237522 | Thank you very much for your answer (I am not familiar enough with the notions in it, so it will take me some time to accept it). Please, there is something in your answer that I do not understand (probably because my lack of knowledge in algebraic geometry): Did you claim that if $w$ is of degree $4$, then there if no endomorphism $f$ of $\mathbb{C}[x,y]$ that fixes $w$? But, for example, if $w=x^4+y^4$ then $\alpha: (x,y) \mapsto (y,x)$ fixes $w$. What am I missing? | |
| Jun 4, 2017 at 3:48 | history | answered | Will Sawin | CC BY-SA 3.0 |