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Difference between revisions of "Abhyankar–Moh theorem"

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An affine [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200101.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200102.png" /> an [[Algebraically closed field|algebraically closed field]] of characteristic zero) is said to have the Abhyankar–Moh property if every imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200103.png" /> extends to an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200104.png" />. The original Abhyankar–Moh theorem states that an imbedded affine line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200105.png" /> has the Abhyankar–Moh property, [[#References|[a1]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200106.png" /> be a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200107.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200108.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a1200109.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001012.png" />, suppose in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001013.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001014.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001015.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001017.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001018.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001019.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001020.png" /> small in comparison with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001021.png" /> and has  "nice"  singularities, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001022.png" /> has the Abhyankar–Moh property [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]]. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001023.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001024.png" />-cross <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001025.png" /> has the Abhyankar–Moh property, [[#References|[a3]]]. The case of a hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001026.png" /> is still open (1998).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001027.png" /> to automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120010/a12001028.png" />"  ''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>
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<table>
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<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>
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<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>
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<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>
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<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>
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<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>
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</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
How to Cite This Entry:
Abhyankar–Moh theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abhyankar%E2%80%93Moh_theorem&oldid=17704
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article