Difference between revisions of "Abhyankar–Moh theorem"
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− | An affine [[ | + | {{TEX|done}} |
+ | An affine [[algebraic variety]] $X \subset k^n$ (with $k$ an [[algebraically closed field]] of [[Characteristic of a field|characteristic]] zero) is said to have the Abhyankar–Moh property if every imbedding $\phi : X \rightarrow k^n$ extends to an automorphism of $k^n$. The original Abhyankar–Moh theorem states that an imbedded affine line in $k^2$ has the Abhyankar–Moh property, [[#References|[a1]]]. | ||
− | The algebraic version of this theorem (which works over any field) is as follows. Let | + | The algebraic version of this theorem (which works over any field) is as follows. Let $k$ be a field of characteristic $p \ge 0$. Let $f,g \in k[T] \setminus k$ be such that $k[f,g] = k[T]$. Let $n = \deg f$ and $m = \deg g$. If $p > 0$, suppose in addition that $p$ does not divide $\mathrm{hcf}(f,g)$. Then $m$ divides $n$ or $n$ divides $m$. |
− | If | + | If $X \subset \mathbb{C}^n$ has $\dim X$ small in comparison with $n$ and has "nice" singularities, then $X$ has the Abhyankar–Moh property [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]]. For every $n$, the $n$-cross $\{x \in \mathbb{C}^n : x_1\cdots x_n = 0 \}$ has the Abhyankar–Moh property, [[#References|[a3]]]. The case of a hyperplane in $\mathbb{C}^n$ is still open (1998). |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.S. Abhyankar, T-t. Moh, "Embeddings of the line in the plane" ''J. Reine Angew. Math.'' , '''276''' (1975) pp. 148–166</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Z. Jelonek, "A note about the extension of polynomial embeddings" ''Bull. Polon. Acad. Sci. Math.'' , '''43''' (1995) pp. 239–244</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Z. Jelonek, "A hypersurface that has the Abhyankar–Moh property" ''Math. Ann.'' , '''308''' (1997) pp. 73–84</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Kalliman, "Extensions of isomrphisms between affine algebraic subvarieties of | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S.S. Abhyankar, T-t. Moh, "Embeddings of the line in the plane" ''J. Reine Angew. Math.'' , '''276''' (1975) pp. 148–166</TD></TR> | ||
+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> Z. Jelonek, "A note about the extension of polynomial embeddings" ''Bull. Polon. Acad. Sci. Math.'' , '''43''' (1995) pp. 239–244</TD></TR> | ||
+ | <TR><TD valign="top">[a3]</TD> <TD valign="top"> Z. Jelonek, "A hypersurface that has the Abhyankar–Moh property" ''Math. Ann.'' , '''308''' (1997) pp. 73–84</TD></TR> | ||
+ | <TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Kalliman, "Extensions of isomrphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$" ''Proc. Amer. Math. Soc.'' , '''113''' (1991) pp. 325–334</TD></TR> | ||
+ | <TR><TD valign="top">[a5]</TD> <TD valign="top"> V. Srinivas, "On the embedding dimension of the affine variety" ''Math. Ann.'' , '''289''' (1991) pp. 125–132</TD></TR> | ||
+ | </table> |
Latest revision as of 19:08, 27 December 2014
An affine algebraic variety $X \subset k^n$ (with $k$ an algebraically closed field of characteristic zero) is said to have the Abhyankar–Moh property if every imbedding $\phi : X \rightarrow k^n$ extends to an automorphism of $k^n$. The original Abhyankar–Moh theorem states that an imbedded affine line in $k^2$ has the Abhyankar–Moh property, [a1].
The algebraic version of this theorem (which works over any field) is as follows. Let $k$ be a field of characteristic $p \ge 0$. Let $f,g \in k[T] \setminus k$ be such that $k[f,g] = k[T]$. Let $n = \deg f$ and $m = \deg g$. If $p > 0$, suppose in addition that $p$ does not divide $\mathrm{hcf}(f,g)$. Then $m$ divides $n$ or $n$ divides $m$.
If $X \subset \mathbb{C}^n$ has $\dim X$ small in comparison with $n$ and has "nice" singularities, then $X$ has the Abhyankar–Moh property [a2], [a4], [a5]. For every $n$, the $n$-cross $\{x \in \mathbb{C}^n : x_1\cdots x_n = 0 \}$ has the Abhyankar–Moh property, [a3]. The case of a hyperplane in $\mathbb{C}^n$ is still open (1998).
References
[a1] | S.S. Abhyankar, T-t. Moh, "Embeddings of the line in the plane" J. Reine Angew. Math. , 276 (1975) pp. 148–166 |
[a2] | Z. Jelonek, "A note about the extension of polynomial embeddings" Bull. Polon. Acad. Sci. Math. , 43 (1995) pp. 239–244 |
[a3] | Z. Jelonek, "A hypersurface that has the Abhyankar–Moh property" Math. Ann. , 308 (1997) pp. 73–84 |
[a4] | S. Kalliman, "Extensions of isomrphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$" Proc. Amer. Math. Soc. , 113 (1991) pp. 325–334 |
[a5] | V. Srinivas, "On the embedding dimension of the affine variety" Math. Ann. , 289 (1991) pp. 125–132 |
Abhyankar–Moh theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abhyankar%E2%80%93Moh_theorem&oldid=17704