Fiche de TP1
Exercice 1
restart;
f := X^8+X^6-3*X^4-3*X^3+8*X^2+2*X-5;
g := 3*X^6+5*X^4-4*X^2-9*X+21;
LDAqJClJIlhHNiIiIikiIiJGKCokKUYlIiInRihGKComIiIkRigpRiUiIiVGKCEiIiomRi1GKClGJUYtRihGMComRidGKClGJSIiI0YoRigqJkY1RihGJUYoRigiIiZGMA==
LCwqJiIiJCIiIilJIlhHNiIiIidGJUYlKiYiIiZGJSlGJyIiJUYlRiUqJkYtRiUpRiciIiNGJSEiIiomIiIqRiVGJ0YlRjEiI0BGJQ==
r:= rem(f,g,X,'q');q;
LCgqJiMiIiYiIioiIiIpSSJYRzYiIiIlRichIiIqJiNGJ0YmRicpRikiIiNGJ0YnI0YnIiIkRiw=
LCYqJiMiIiIiIiRGJSlJIlhHNiIiIiNGJUYlI0YqIiIqISIi
evalb(f=expand(g*q+r));
SSV0cnVlRyUqcHJvdGVjdGVkRw==
Delta:=gcdex(f,g,X,'A','B');A;B;
IiIi
LC4qJiMiJiopUiIiJ2EuOCIiIilJIlhHNiIiIiZGJ0YnKiYjIiVEIyoiJnheJ0YnKUYpIiIlRidGJyomIyImIkc/Ri9GJylGKSIiJEYnRicqJiMiJkRyJ0YmRicpRikiIiNGJ0YnKiYjIiVcXkYmRidGKUYnRicjIiUiUiIiJkEnPSEiIg==
LDIqJiMiJWpZIidhLjgiIiIpSSJYRzYiIiIoRichIiIqJiMiJXZJIiZ4XidGJylGKSIiJ0YnRiwqJiMiJTFfRjBGJylGKSIiJkYnRiwqJiMiJnYjPUYmRicpRikiIiVGJ0YsKiYjIiVXXEYwRicpRikiIiRGJ0YnKiYjIiZ6OiNGJkYnKUYpIiIjRidGJyomIyIlNT5GMEYnRilGJ0YnIyIlKilRRiZGJw==
evalb(Delta=expand(A*f+B*g));
SSV0cnVlRyUqcHJvdGVjdGVkRw==
R:=resultant(f,g,X);
IiczMkU=
a,b:= R*A,R*B;
NiQsLiomIiZ5eiMiIiIpSSJYRzYiIiImRiZGJiomIiYrcCRGJilGKCIiJUYmRiYqJiImQzYpRiYpRigiIiRGJkYmKiYiJ11VOEYmKUYoIiIjRiZGJiomIiYpSDVGJkYoRiZGJiImdSU+ISIiLDIqJiIlRSQqRiYpRigiIihGJkY6KiYiJitCIkYmKUYoIiInRiZGOiomIiZDMyNGJkYnRiZGOiomIiZdbCRGJkYtRiZGOiomIiZ3KD5GJkYxRiZGJiomIiZlSiVGJkY1RiZGJiomIiVTd0YmRihGJkYmIiV5eEYm
evalb(R=expand(a*f+b*g));
SSV0cnVlRyUqcHJvdGVjdGVkRw==
Exercice 2
restart;
f:=t*X^2+3*X-1;
LCgqJilJIlhHNiIiIiMiIiJJInRHRiZGKEYoKiYiIiRGKEYlRihGKEYoISIi
g:=6*X^2+t^2-4;
LCgqJiIiJyIiIilJIlhHNiIiIiNGJUYlKiQpSSJ0R0YoRilGJUYlIiIlISIi
R:=sort(resultant(f,g,X),t);
LC4qJClJInRHNiIiIiciIiJGKComIiIpRigpRiUiIiVGKCEiIiomIiM3RigpRiUiIiRGKEYoKiYiI3FGKClGJSIiI0YoRigqJiIjW0YoRiVGKEYtIiQhPUYt
f0:=subs(t=0,f):g0:=subs(t=0,g): #on sp\303\251cialise en t=0, le degr\303\251 de f chute de p=1
R0:=resultant(f0,g0,X);subs(t=0,R); # R(0)=lcoeff(g,X)^p*R0
ISNJ
ISQhPQ==
restart;
f:=t*X^2+3*t*X-1;
LCgqJilJIlhHNiIiIiMiIiJJInRHRiZGKEYoKigiIiRGKEYlRihGKUYoRihGKCEiIg==
g:=6*X^2+t^2-4;
LCgqJiIiJyIiIilJIlhHNiIiIiNGJUYlKiQpSSJ0R0YoRilGJUYlIiIlISIi
R:=sort(resultant(f,g,X),t);
LC4qJClJInRHNiIiIiciIiJGKComIiNZRigpRiUiIiVGKEYoKiYiIzdGKClGJSIiJEYoRigqJiIkKyNGKClGJSIiI0YoISIiKiYiI1tGKEYlRihGNSIjT0Yo
f0:=subs(t=0,f):g0:=subs(t=0,g): #on sp\303\251cialise en t=0, le degr\303\251 de f chute de p=2
R0:=resultant(f0,g0,X);subs(t=0,R); # R(0)=lcoeff(g,X)^p*R0
IiIi
IiNP
restart;
f:=t*X^2+X-1;
LCgqJilJIlhHNiIiIiMiIiJJInRHRiZGKEYoRiVGKEYoISIi
g:=t*X^2+X+t^2-4;
LCoqJilJIlhHNiIiIiMiIiJJInRHRiZGKEYoKiQpRilGJ0YoRihGJUYoIiIlISIi
R:=sort(resultant(f,g,X),t);
KiYpLCYqJClJInRHNiIiIiMiIiJGKiIiJCEiIkYpRipGJkYq
f0:=subs(t=0,f):g0:=subs(t=0,g): #on sp\303\251cialise en t=0, les degr\303\251s de f rt de g chutent
R0:=resultant(f0,g0,X);subs(t=0,R);
ISIk
IiIh
Exercice 3
restart;
with(plots):
f:=X^4+Y^4-1;
LCgqJEkiWEc2IiIiJSIiIiokSSJZR0YlRiZGJyEiIkYn
g:=X^5*Y^2-4*X^3*Y^3+X^2*Y^5-1;
LCoqJkkiWEc2IiIiJkkiWUdGJSIiIyIiIiomRiRGKEYnRiZGKSomRiQiIiRGJ0YsISIlISIiRik=
D1:=implicitplot(f,X=-3..3,Y=-3..3,grid=[200,200],color=blue):
D2:=implicitplot(g,X=-3..3,Y=-3..3,grid=[200,200],color=red):
display({D1,D2});
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R:=resultant(f,g,Y); # le degr\303\251 de f en Y ne chute pas: \303\251quation de la projection
# de l'intersection des courbes sur l'axe des X
LE4qJEkiWEc2IiIjRyIiIyokRiQiI0YhIzsqJEYkIiNFIiNLKiRGJCIjQyIkXCMqJEYkIiNCIiNbKiRGJCIjQSEkRyIqJEYkIiNAIiIlKiRGJCIjPyEkZCgqJEYkIiM+ISQ3IiokRiQiIz0iJCM+KiRGJCIjPCEjNyokRiQiIzsiJGUoKiRGJCIjOiIkVyIqJEYkIiM5ISRFIiokRiQiIzhGJiokRiQiIzchJF4jKiRGJCIjNiEjayokRiQiIzUiI0kqJEYkIiIqISNPKiRGJCIiKSEiIiokRiQiIiZGRyIiIkZcbw==
x:=fsolve(R,X); # les ordonn\303\251es approch\303\251es des points d'intersection
NiYkISskbyc0VSMqISM1JCErcSkqR3VmRiUkIitpUTg2c0YlJCIrcEgxbCcqRiU=
# il y a sym\303\251trie de l'intersection par rapport \303\240 la 1ere bissectrice puisque
# les polyn\303\264mes f et g sont sym\303\251triques.
with(combinat, cartprod):
pt:=[]:T:=cartprod([[x], [x]]):
while not T[finished] do
a,b:=op(T[nextvalue]());
if abs(subs({X=a,Y=b},f))<10^(-7) and abs(subs({X=a,Y=b},g))<10^(-8) then
pt:=[op(pt),[a,b]] end if;
end do:
pt; # les coordonn\303\251es approch\303\251es des points d'intersection
NyY3JCQhKyRvJzRVIyohIzUkIitpUTg2c0YmNyQkIStxKSpHdWZGJiQiK3BIMWwnKkYmNyRGJ0YkNyRGLEYq