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version 1.3, 2001/09/25 01:17:08 version 1.4, 2001/09/25 02:28:27
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 %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.2 2001/09/23 08:31:18 takayama Exp $  %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.3 2001/09/25 01:17:08 takayama Exp $
 \documentclass{slides}  \documentclass{slides}
 %%\documentclass[12pt]{article}  %%\documentclass[12pt]{article}
 \usepackage{color}  \usepackage{color}
Line 141  ox_reset(P);
Line 141  ox_reset(P);
 \noindent  \noindent
 {\color{green} Example 1} \\  {\color{green} Example 1} \\
 Theorem (Cantor-Zassenhaus) \\  Theorem (Cantor-Zassenhaus) \\
 Let $f_1$ and $f_2$ be degree $d$ polynomials in $F_q[x]$.  Let $f_1$ and $f_2$ be degree $d$ irreducible polynomials in $F_q[x]$.
 For a random degree $2d-1$ polynomial $g \in F_q[x]$,  For a random degree $2d-1$ polynomial $g \in F_q[x]$,
 the chance of  the chance of
 $$ GCD(g^{(q^d-1)/2}-1,f_1 f_2) = f_1 \,\mbox{or}\, f_2 $$  $$ GCD(g^{(q^d-1)/2}-1,f_1 f_2) = f_1 \,\mbox{or}\, f_2 $$
Line 205  def c_z(F,E,Level)
Line 205  def c_z(F,E,Level)
 \end{verbatim}  \end{verbatim}
 \newpage  \newpage
   
   \epsfxsize=17cm
   \epsffile{cz.ps}
   
   \noindent
   {\color{blue} Performance of parallel CZ algorithm} \\
   $d=1$, $k=200$ : product of $200$ linear forms. \\
   $d=2$, $k=50$ : product of $50$ irreducible degree $2$ polynomials. \\
   
   \newpage
 {\color{green} Example 2} \\  {\color{green} Example 2} \\
 Shoup's algorithm to multyply polynomials.  Shoup's algorithm to multiply polynomials.  \\
   {\color{green} Example 3} \\
   Competitive Gr\"obner basis computation. \\
 \newpage  \newpage
   
 \noindent  \noindent
 {\color{red} 5. e-Bateman project} \\  {\color{red} 5. e-Bateman project} (Electronic mathematical formula book)\\
 First Step: \\  First Step: \\
 Gauss Hypergeometric function:  Gauss Hypergeometric function:
 $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty  $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty
   \frac{(a)_n (b)_n}{(1)_n}{(c)_n} x^n    \frac{(a)_n (b)_n}{(1)_n (c)_n} x^n
 $$  $$
 where  where
 $$ (a)_n = a(a+1) \cdots (a+n-1). $$  $$ (a)_n = a(a+1) \cdots (a+n-1). $$
   {\color{green}
 $$ \log (1+x) = x F(1,1,2;-x) $$  $$ \log (1+x) = x F(1,1,2;-x) $$
 $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$  $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$
   }
   
 \noindent  \noindent
 Appell's $F_1$:  Appell's $F_1$:
Line 270  for GKZ hypergeometric systems.
Line 283  for GKZ hypergeometric systems.
   
 \newpage  \newpage
 \noindent  \noindent
 {\color{green} Competitive Gr\"obner Basis Computation}  {\color{green} Example 3. Competitive Gr\"obner Basis Computation}
 \begin{verbatim}  \begin{verbatim}
 extern Proc1,Proc2$  extern Proc1,Proc2$
 Proc1 = -1$ Proc2 = -1$  Proc1 = -1$ Proc2 = -1$
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