TP8
Exercice 1.
restart;
with(plots):
with(algcurves):
F:=X^2*Y^5-X^5*Y^2-2*X*Y^5+X^5+Y^5-Y*X^3-X^2*Y^2-Y^3*X;
LDIqJkkiWEc2IiIiJkkiWUdGJSIiIyEiIiomRiRGKEYnRiYiIiIqJkYkRitGJ0YmISIjKiRGJEYmRisqJEYnRiZGKyomRiQiIiRGJ0YrRikqJkYkRihGJ0YoRikqJkYkRitGJ0YxRik=
G:=implicitplot(F,X=-6..5,Y=-6..5,grid=[250,250],color=blue,thickness=1):
display(G);
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2(>#3?(f+'>$!#;$"0CN$=;R`T!#97$$"1<Or.ue/K!#:$"*dU1;%!")7$$"1ZU))H.E?K!#:$"04j#oRJtT!#97$$"+F<$HB$!"*$"1&4`l[aI=%!#:7$$"2y@#zwB_WK!#;$"2:n?%H'GK>%!#;7$$"1^`Qp%H$eK!#:$"*G>[?%!")7$$"28KxR)>uoK!#;$"1wppGd=8U!#:7$$"+M%3rF$!"*$"1(euIf&e>U!#:7$$"2n!*p&*4AGH$!#;$"1&**>pJ#GLU!#:7$$"1hey*p^;J$!#:$"*)f**[U!")7$$"2ZJ1$[l+<L!#;$"1UU)HduKD%!#:7$$"+T^G@L!"*$"0:CpBLlD%!#97$$"1$[KD!*>4M$!#:$"2&QT8Fz`tU!#;7$$"2x"GD"yuXO$!#;$"*psJH%!")7$$"2ANWS(*o]O$!#;$"1FB)RclNH%!#:7$$"+[=YlL!"*$"1&H9GjjQH%!#:7$$"1L9U!HH))Q$!#:$"1/<*f&>)RJ%!#:7$$"+b&Q'4M!"*$"1WZ&3%3<JV!#:7$$"1W*)z:4%GT$!#:$"1'e$)*Qq9MV!#:7$$"2a.aF2:kT$!#;$")%\tL%!"(7$$"21")Q)QEcOM!#;$"/GLC?gaV!#87$$"+i_"QX$!"*$"1EP)*\.zoV!#:7$$"2BUf0'eVgM!#;$"186%3^0\P%!#:7$$"2X'R$\]kyY$!#;$"*6E:Q%!")7$$"1o<Y288%[$!#:$"1-dlonQ&R%!#:7$$"+o>*z\$!"*$"1"e0U?SnS%!#:7$$"20HWLhwy]$!#;$"1w>Am"=eT%!#:7$$"2'ev#*\?->N!#;$"*"GqDW!")7$$"2aUM:uX:`$!#;$"1op=WdKOW!#:7$$"+v'o@a$!"*$"1YGa4e*\W%!#:7$$"1m3R5F<bN!#:$"2D/EP[voX%!#;7$$"2M)>V8:!*pN!#;$"*_z)pW!")7$$"2WigBc:)yN!#;$"1>2:S$4uZ%!#:7$$"+#QXje$!"*$"17")*e'\`$[%!#:7$$"1fH:SHL-O!#:$"0Mh2no!)\%!#97$$"2i^8s,;0i$!#;$"*BcS^%!")7$$"2%yPII'\fi$!#;$"2NprzoG'=X!#;7$$"+*3A0j$!"*$"2l(Ga$ePB_%!#;7$$"2C8D'>aO\O!#;$"1z1O-'*QRX!#:7$$"20p!z^!y3n$!#;$"*$HBeX!")7$$"1dACDh&Hn$!#:$"2a3w3gv*fX!#;7$$"+'z)puO!"*$"1u%zqV&QhX!#:7$$"1A3:lwF'p$!#:$"1xPQp2$3e%!#:7$$"+.b()=P!"*$"1R(*QZcb+Y!#:7$$"205;/_=)>P!#;$"19&)\AmY,Y!#:7$$"13S]QD#3s$!#:$"*k4Cg%!")7$$"2OLTbjwIu$!#;$"1ktE:_QAY!#:7$$"+4A0jP!"*$"1Hfj)Ql(RY!#:7$$"2F**flYHlw$!#;$"0(Q?#44Jk%!#97$$"0*QaokCqP!#9$"*N'eYY!")7$$"2&yIuy(o(*y$!#;$"1$G!\%[YSm%!#:7$$"+;*Gs!Q!"*$"1nB%pT$>zY!#:7$$"2.(f?R%RJ"Q!#;$"1z0tFD&[o%!#:7$$"1y&=!35[>Q!#:$"*0j2p%!")7$$"1:^;'3gj$Q!#:$"19m&ye3eq%!#:7$$"+BcS^Q!"*$"1iApcc#)=Z!#:7$$"2B,@@)RlfQ!#;$"1\)=.S"pEZ!#:7$$"2:oPseM&oQ!#;$"*wR\t%!")7$$"2sLB#Hh&G)Q!#;$"2l)3khfmZZ!#;7$$"*L#e&*Q!")$"1)e7=A\'eZ!#:7$$"1TR/Y#yg!R!#:$"1"HVKb?'oZ!#:7$$"02iy>:u"R!#9$"*Z;"zZ!")7$$"2C&*zS3i#HR!#;$"1enD@Mh*y%!#:7$$"+P!f(RR!"*$"1')He!H_')z%!#:7$$"2cTb+.<C&R!#;$"1,.&*y^j5[!#:7$$"2%=>+V/8mR!#;$"*<$HB[!")7$$"2$379vFevR!#;$"1MgURhkJ[!#:7$$"+Wd$R)R!"*$"1_Te(*R#)Q[!#:7$$"2k<[;#[n)*R!#;$"1ns,,3t_[!#:7$$"1yR8)e(o9S!#:$"*))pu'[!")7$$"1Vrl3F#=-%!#:$"1lB">ifP([!#:7$$"*X7"GS!")$"1V=?RV:z[!#:7$$"1N1*e!e&[/%!#:$"1l$4TB.\*[!#:7$$"1knk6N4jS!#:$"*fY;"\!")7$$"06mG3')z1%!#9$"0$)fhm\f"\!#97$$"*;*GsS!")$"1/VX%4M'>\!#:7$$"1[P*)*)Q'44%!#:$"1_i5g&[r$\!#:7$$"1R)Gava86%!#:$"*HBe&\!")7$$"1NPt7o29T!#:$"2v02yL7#e\!#;7$$"*'eY;T!")$"1,3^GZDg\!#:7$$"1x**\QF+PT!#:$"1B+]@JYz\!#:7$$"1kZ4tvZfT!#:$"#]!""-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%*THICKNESSG6#"""-&%&_AXISG6#"""6'-%+_GRIDLINESG6'-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6'-%+_GRIDLINESG6'-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%)_VISIBLEG6#""!-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"X6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"Y6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$I"!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%?Q!""-%.BOUNDS_HEIGHTG6#$"%?Q!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"G6"
S:=solve({F,diff(F,X),diff(F,Y)},{X,Y});
PCQvSSJYRzYiIiIhL0kiWUdGJUYm
l'origine est le seul point singulier
for m from 0 to infinity do
CT:=mtaylor(F,{X=0,Y=0},m+1);
if CT<>0 then break end if;
end do:
CT;
LCgqJkkiWEc2IiIiJEkiWUdGJSIiIiEiIiomRiQiIiNGJ0YrRikqJkYkRihGJ0YmRik=
op({allvalues(evala(AFactor(CT)))});
LCQqKkkiWEc2IiIiIiwmRiRGJiomLCYqJl4jIyEiIiIiI0YmIiIkI0YmRi5GJkYwRiZGJkkiWUdGJUYmRiZGJiwmRiRGJiomLCYqJl4jRjBGJkYvRjBGJkYwRiZGJkYxRiZGJkYmRjFGJkYt
l'origine est donc un point multiple d'ordre 4 ordinaire
Exercice 2
restart;
with(LinearAlgebra):
M:=<<6,12,18>|<8,12,4>|<4,18,4>|<20,30,10>>;
LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNXUmW2RMdVN1WSU9
U,S,V:=SmithForm(M,output=['U','S','V']);
NiUtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1OTd2TlYyV25XPS1GJDYjL0YrIjUlNF5kTHVTdVklPS1GJDYjL0YrIjVNOHZOVjJXblc9
Equal(S,U.M.V);
SSV0cnVlRyUqcHJvdGVjdGVkRw==
nl,nc:=Dimension(M);r:=min(nl,nc);
NiQiIiQiIiU=
IiIk
with(combinat,choose):
for k from 1 to r do
CL:=choose(nl,k);
CC:=choose(nc,k);
SM[k]:=[seq(seq(M[a,b],a=CL),b=CC)];
end do:
SM[1];
Ny4tSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1JXBeZEx1U3VZJT0tRiQ2Iy9GKyI1OT12TlYyV25XPS1GJDYjL0YrIjVNPnZOVjJXblc9LUYkNiMvRisiNWE/dk5WMlduVz0tRiQ2Iy9GKyI1JTRbZEx1U3VZJT0tRiQ2Iy9GKyI1OSNbZEx1U3VZJT0tRiQ2Iy9GKyI1TSRbZEx1U3VZJT0tRiQ2Iy9GKyI1YSVbZEx1U3VZJT0tRiQ2Iy9GKyI1OSlbZEx1U3VZJT0tRiQ2Iy9GKyI1YSFcZEx1U3VZJT0tRiQ2Iy9GKyI1dSJcZEx1U3VZJT0tRiQ2Iy9GKyI1JUhcZEx1U3VZJT0=
SM[2];
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
SM[3];
NyYtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1YSU+alB1U3VZJT0tRiQ2Iy9GKyI1dSY+alB1U3VZJT0tRiQ2Iy9GKyI1JXA+alB1U3VZJT0tRiQ2Iy9GKyI1OSk+alB1U3VZJT0=
for k from 1 to r do
DM[k]:=map(Determinant,SM[k]);
end do:
DM[1]; # mineurs d'ordre 1
Ny4iIiciIzciIz0iIilGJCIiJUYnRiVGJyIjPyIjSSIjNQ==
DM[2]; # mineurs d'ordre 2
NzQhI0MhJD8iISRvIiIjZyEjWyEkdyMhI2chJCskISQ/JSIjJyoiIztGIyIiIUYuRi4hJFMjISNTRiY=
DM[3]; #mineurs d'ordre 3
NyYiJSNSIiIiISElIVskRiQ=
for k from 1 to r do
d[k]:= igcd(op(DM[k]));
end do:
d[1];d[2];d[3]; #les invariants d\357\277\275terminantiels
IiIj
IiIl
IiQncA==
finv:=[d[1],seq(d[k]/d[k-1],k=2..r)]; # les facteurs invariants
NyUiIiNGIyIkdSI=
Uinv:=U^(-1);
LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNU0wS3dWMlduVz0=
E:=[seq(Column(Uinv,j),j=1..ColumnDimension(Uinv))];
NyUtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIjVhMUt3VjJXblc9LUYkNiMvRi4iNXUyS3dWMlduVz0tRiQ2Iy9GLiI1JSozS3dWMlduVz0=
U.E[1],U.E[2],U.E[3];;
NiUtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIjU5NUt3VjJXblc9LUYkNiMvRi4iNU02S3dWMlduVz0tRiQ2Iy9GLiI1YTdLd1YyV25XPQ==
F:=[seq(finv[j]*E[j],j=1..ColumnDimension(Uinv))];
NyUtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIjU5O0t3VjJXblc9LUYkNiMvRi4iNWE9S3dWMlduVz0tRiQ2Iy9GLiI1JTRBalB1U3VZJT0=
Column(M,1);
LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1OUFLd1YyV25XPQ==
Exercice 3.
restart;
with(LinearAlgebra):
A:=<<1,0,1,0,0,1>|<-2,1,-5/2,-1,0,-3/2>|<4,5/2,2,9/2,2,-1/2>|<-2,-7/2,-1/2,-7/2,-2,1>|<5,2,5/2,3,3,3/2>|<-4,-5/2,-3,-7/2,-1,1/2>>;
LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNTFdQXdWMlduVz0=
n,n:=Dimension(A);
NiQiIidGIw==
P:=sort(CharacteristicPolynomial(A,X),X);
LC4qJEkiWEc2IiIiJyIiIiokRiQiIiYhIiUqJEYkIiIkIiM7KiRGJCIiIyEjN0YkISM7Ri1GJw==
Q:=sort(MinimalPolynomial(A,X),X); # le polyn^ome minimal est un invariant de similitude
LC4qJEkiWEc2IiIiJiIiIiokRiQiIiUhIiMqJEYkIiIkISIlKiRGJCIiIyIiKUYkRikhIilGJw==
P est le produit des invariants de similitude et Q est le plus grand (pour la relation de divisibilit\303\251) or deg(P)=n, deg(Q)=n-1 il ne peux donc y avoir qu'un seul autre invariant de degr\303\251 1 qui est n\303\251cessairement P/Q
sort(quo(P,Q,X),X); # il n'y a qu'un autre invariant de similitude
LCZJIlhHNiIiIiIhIiNGJQ==
M:=(-1)^n*CharacteristicMatrix(A,X);
LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNUUiR2lQdVN1WSU9
S,U,V:=SmithForm(M,X,output=['S','U','V']):
S:=map(sort,S,X);
LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNUk7QXdWMlduVz0=
Equal(S,map(expand,U.M.V));
SSV0cnVlRyUqcHJvdGVjdGVkRw==
Fi:=remove(type,[seq(S[n-i,n-i],i=0..n-1)],constant); # la liste les invariants de similitude
NyQsLiokSSJYRzYiIiImIiIiKiRGJSIiJSEiIyokRiUiIiQhIiUqJEYlIiIjIiIpRiVGKiEiKUYoLCZGJUYoRitGKA==
TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzM1NzQ4MDk0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjW28iKSIpIiIiRiZGJkYmRiZGJkYmRiZGJiIiI0YmRidGJkYnRiZGJ0YmRiYiIiRGJkYmRihGJkYmRiZGJ0YmIiIlRiZGJ0YmRilGJkYmRiZGJiIiJkYmRiZGJkYmRidGKEYnRiYiIidGJkYnRiZGJkYmRiZGJkYmIiIoRiZGJkYnRiZGKUYmRidGJiIiKUYl TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc2MjI1MDA2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjRSInIiciIiIiIiFGJkYnRidGJiEiI0YmIyEiJiIiIyEiIkYnIyEiJEYrIiIlIyIiJkYrRisjIiIqRitGKyNGLEYrRigjISIoRitGNEY1RihGJkYxRitGMCIiJEY3I0Y3RishIiVGKUYuRjVGLCNGJkYrRiU= TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc2MjI4MTI2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjRSInIicsJiUiWEciIiIhIiJGKCIiIUYpRipGKkYpIiIjRiYjIiImRitGKEYqIyIiJEYrISIlIyEiJkYrLCZGJ0YoISIjRigjISIqRitGNCNGKEYrRisjIiIoRitGNywmRidGKEY4RihGK0YpRjJGNEYxISIkLCZGJ0YoRjtGKCNGO0YrIiIlRixGL0Y4RigsJkYnRigjRilGK0YoRiU= TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Mzc2MjIxNjMwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjRSInIiciIiIiIiFGJ0YnRidGJ0YnRiZGJ0YnRidGJ0YnRidGJkYnRidGJ0YnRidGJ0YmRidGJ0YnRidGJ0YnLCYlIlhHRiYhIiNGJkYnRidGJ0YnRidGJ3ovNiNGKSZGJiVGKiQhIiUjIiIpIiIiJSEhIilGJQ==