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Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

Wednesday, July 14, 14:50 ~ 15:40 UTC-3

A note on the Jacobian conjecture

Zbigniew Jelonek

Polska Akademia Nauk, Poland   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9cf39ea3d6427acc6f3670c55662b5fa').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9cf39ea3d6427acc6f3670c55662b5fa = 'n&#97;j&#101;l&#111;n&#101;' + '&#64;'; addy9cf39ea3d6427acc6f3670c55662b5fa = addy9cf39ea3d6427acc6f3670c55662b5fa + 'cyf-kr' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'pl'; var addy_text9cf39ea3d6427acc6f3670c55662b5fa = 'n&#97;j&#101;l&#111;n&#101;' + '&#64;' + 'cyf-kr' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'pl';document.getElementById('cloak9cf39ea3d6427acc6f3670c55662b5fa').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9cf39ea3d6427acc6f3670c55662b5fa + '\'>'+addy_text9cf39ea3d6427acc6f3670c55662b5fa+'<\/a>';

Let $F:\mathbb{C}^n\to\mathbb{C}^n$ be a polynomial mapping with a non-vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, if $S_F$ is connected, then $\chi(S_F)>0.$ Additionally, if $n=2$, then the set $S_F$ of non-properness of $F$ cannot be a curve without self-intersections.