hojas jupyter

Mult3.tex

@@ -307,7 +307,10 @@
 We propose the following general method to study germs of multiplicity 3 singular points of hypersurfaces. Firstly, consider an equation of the germ given by a monic polynomial $F = z^3 + a_1z + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. By means of a Tschirnhaus transformation, reduce $F$ to a monic equation $z^3 + b_2z + b_3$ that does not have terms of degree 2 with respect to $z$. Secondly, consider an embedded resolution $\pi: X \rightarrow \CC^d$ of the reduced divisor given by the product $b_2b_3$ of the coefficients of degree one and zero with respect to $z$. Consider the usual stratification of the pair $(X, (\pi \circ F)^{-1} \{F=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ F)^{-1} \{F=0\}$ into irreducible components. Pulling back $F$ under the map $\pi \times id$, we obtain for each stratum $E_I^\circ$ a quasi-trinomial local expression for $F$.
 The local quasi-trinomial expression will be non degenerate with respect to its Newton polyhedron unless the monomials $M_1$, $M_2$ and their coefficients $c_1$ and $c_2$ satisfy a precise condition.
 
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 \section{The quasi-trinomial case}
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 In this section we consider the case of
 $$F = z^3 + M_1(c_1+A)z + M_2(c_2 + M_2) \in \CC[[{\bf x}]][z]$$
 where $M_1={\bf x}^{m_1}$ and $M_2={\bf x}^{m_2}$ are monomials, $c_1,c_2 \in \CC^\ast$ and $A, B \in ({\bf x})\CC[[{\bf x}]]$
@@ -315,7 +318,9 @@ where $M_1={\bf x}^{m_1}$ and $M_2={\bf x}^{m_2}$ are monomials, $c_1,c_2 \in \C
 We distinguish the following two cases
 \begin{itemize}
 
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 \item Assume that the monomials $M_1$ and $M_2$ are not related by the divisibility relation, i.e., neither $M_1 \nmid M_2$ nor $M_1 \nmid M_2$.
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 In this case, the Newton polyhedra of $F$ contains a unique two dimensional compact face determined by the monomials $z^3$, $M_1z$ and $M_2$. Moreover, $F$ is non degenerate with respect to its Newton polyhedra.
 
 \
@@ -346,6 +351,7 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \fill [yellow] (11,0,4)--(11,0,0)--(13,0,0)--(13,0,4)--(11,0,4);
 \fill [yellow] (11,0,4)--(11,0,7)--(6,0,7)--(6,0,4)--(11,0,4);
 
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 \draw[gray, thick] (6,0,0) -- (6,0,7);
 \draw[gray, thick] (6,0,0) -- (6,4,0);
 \draw[gray, thick] (6,0,0) -- (13,0,0);
@@ -384,7 +390,9 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \filldraw[black] (9,0,6) circle (2pt) node[anchor=north]{$M_1$};
 \filldraw[black] (9,1,6) circle (2pt) node[anchor=south west]{$M_1z$};
 
-}
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+
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 \end{tikzpicture}
 
 \
@@ -440,9 +448,11 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 
 
 
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 \fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
 
 
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 \draw[gray, thick] (6,0,0) -- (6,0,7);
 \draw[gray, thick] (6,0,0) -- (6,4,0);
 \draw[gray, thick] (6,0,0) -- (13,0,0);
@@ -512,6 +522,7 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
 
 
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 \draw[gray, thick] (6,0,0) -- (6,0,7);
 \draw[gray, thick] (6,0,0) -- (6,4,0);
 \draw[gray, thick] (6,0,0) -- (13,0,0);
@@ -570,9 +581,11 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 \begin{tikzpicture}
 
 
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 \fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
 
 
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 \draw[gray, thick] (6,0,0) -- (6,0,7);
 \draw[gray, thick] (6,0,0) -- (6,4,0);
 \draw[gray, thick] (6,0,0) -- (13,0,0);
@@ -592,7 +605,9 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 \draw[gray, thick] (6,3,0) -- (13,3,0);
 
 
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 \draw[gray, dashed] (6,1,8/3) -- (6,0,8/3);
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 \draw[gray, dashed] (28/3,0,0) -- (28/3,0,8/3);
 \draw[gray, dashed] (6,1,8/3) -- (6,1,0);
 \draw[gray, dashed] (28/3,1,0) -- (28/3,0,0);