TP7 Exercice 1 restart: with(Groebner);

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n:=8: A:=[[1,2],[1,5],[1,6],[2,3],[2,4],[2,8],[3,4],[3,8],[4,5], [4,7],[5,6],[5,7],[6,7],[7,8]];

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F0:=seq(X[i]^3-1,i=1..n);

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F1:=seq(X[a[1]]^2+X[a[1]]*X[a[2]]+X[a[2]]^2,a in A);

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F:=[F0,F1];

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G:=Basis(F,plex(seq(X[i],i=1..n)));

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map(g->LeadingMonomial(g,plex(seq(X[i],i=1..n))),G);

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IsZeroDimensional(F);

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for i from 1 to n do p[i]:=UnivariatePolynomial(X[i],F) end do;

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sol:=[solve({op(G)},{seq(X[i],i=1..n)})];

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toutessol:=map(allvalues,sol);

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nops(toutessol);

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Exercice 2 restart; with(Groebner): f1:=X^2*Y-Y+Z;

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f2:=X*Y^2-X+Z;

LCgqJkkiWEc2IiIiIkkiWUdGJSIiI0YmRiQhIiJJIlpHRiVGJg==

f3:=X+Y+Z-1;

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on calcule la base de Gr\303\266bner r\303\251duite pour l'ordre lexicographique G:=Basis([f1,f2,f3],plex(X,Y,Z));

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puis les mon\303\264mes dominants de chaque \303\251l\303\251ment de G lmG:=map(e->LeadingMonomial(e,plex(X,Y,Z)),G);

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les mon\303\264mes standard sont les mon\303\264mes qui ne sont divisilbes par aucub des \303\251l\303\251ments de lmG de sorte que l'on a: B := [1, Y, Z, Z^2, Z^3];

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ces mon\303\264mes forment une base du quotient Q[X,Y,Z]/I qui esr donc de dimension 5 u_ := X^5+Y^5+Z^5-5*X^2*Y-5*Y^2*Z-5*Z*X^2-1;

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u:=NormalForm(u_,G,plex(X,Y,Z));

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u est inversible si et seulement s'il existe v avec uv=1; de plus si v existe il est unique. on cherche v avec des coefficients ind\303\251termin\303\251s. x_1,x_2,x_3,x_3,x_5 on travaille donc dans l'extension Q(x_1,x_2,x_3,x_3,x_5) ; unassign('x'):v:=add(x[i]*B[i],i=1..nops(B));

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p:=NormalForm(expand(u*v)-1,G,plex(X,Y,Z));;

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sol:=solve({coeffs(p,{X,Y,Z})},{seq(x[i],i=1..nops(B))});

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assign(sol);v;

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sur cet exemple i lest possible de r\303\251soudre explicitement le syst\303\250me S1:=[solve({f1,f2,f3},{X,Y,Z})]: S2:=map(allvalues,S1): S3:=map(evalc,S2);

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nops(S3);

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