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Some Reductions on Jacobian Problem in Two Variables | | Wenhua Zhao
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19 Sep 2002 | | Subject: | Algebraic Geometry; Complex Variables; Commutative Algebra MSC-class: 14R15, 14C17 | math.AG math.AC math.CV | | Abstract: | Let $f=(f_1, f_2)$ be a regular sequence of affine curves in $C^2$. Under some reduction conditions achieved by composing with some polynomial automorphisms of $C^2$, we show that the intersection number of curves $(f_i)$ in $C^2$ equals to the coefficient of the leading term $x^{n-1}$ in $g_2$, where $n=deg f_i$ $(i=1, 2)$ and $(g_1, g_2)$ is the unique solution of the equation $y{mathcal J}(f)=g_1f_1+g_2f_2$ with $deg g_ileq n-1$. So the well-known Jacobian problem is reduced to solving the equation above. Furthermore, by using the result above, we show that the Jacobian problem can also be reduced to a special family of polynomial maps. | | Source: | arXiv, math.AG/0209254 | | Services: | Forum | Review | PDF | Favorites | |
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