Résultant/Exercices/Matrice de Sylvester

Exercice 3-1

Soient $P(X)=aX^{4}+bX^{3}+cX^{2}+dX+e$, de degré 4 et $Q(X)=pX^{3}+qX^{2}+rX+s$, de degré 3.

Calculer $\operatorname {Res} (P,Q)$ par un déterminant de Sylvester.

Solution

${\begin{aligned}\operatorname {Res} (P,Q)&={\begin{vmatrix}a&0&0&p&0&0&0\\b&a&0&q&p&0&0\\c&b&a&r&q&p&0\\d&c&b&s&r&q&p\\e&d&c&0&s&r&q\\0&e&d&0&0&s&r\\0&0&e&0&0&0&s\end{vmatrix}}\\&\\&=a{\begin{vmatrix}a&0&q&p&0&0\\b&a&r&q&p&0\\c&b&s&r&q&p\\d&c&0&s&r&q\\e&d&0&0&s&r\\0&e&0&0&0&s\end{vmatrix}}-p{\begin{vmatrix}b&a&0&p&0&0\\c&b&a&q&p&0\\d&c&b&r&q&p\\e&d&c&s&r&q\\0&e&d&0&s&r\\0&0&e&0&0&s\end{vmatrix}}\\&\\&=aeA+asB+peC-psD\end{aligned}}$

avec

$A={\begin{vmatrix}a&q&p&0&0\\b&r&q&p&0\\c&s&r&q&p\\d&0&s&r&q\\e&0&0&s&r\end{vmatrix}},\quad B={\begin{vmatrix}a&0&q&p&0\\b&a&r&q&p\\c&b&s&r&q\\d&c&0&s&r\\e&d&0&0&s\end{vmatrix}},\quad C={\begin{vmatrix}b&a&p&0&0\\c&b&q&p&0\\d&c&r&q&p\\e&d&s&r&q\\0&e&0&s&r\end{vmatrix}},\quad D={\begin{vmatrix}b&a&0&p&0\\c&b&a&q&p\\d&c&b&r&q\\e&d&c&s&r\\0&e&d&0&s\end{vmatrix}}$.

Calcul de $A$

$A=aA_{1}-qA_{2}+pA_{3}$ avec

$A_{1}={\begin{vmatrix}r&q&p&0\\s&r&q&p\\0&s&r&q\\0&0&s&r\end{vmatrix}},\quad A_{2}={\begin{vmatrix}b&q&p&0\\c&r&q&p\\d&s&r&q\\e&0&s&r\end{vmatrix}},\quad A_{3}={\begin{vmatrix}b&r&p&0\\c&s&q&p\\d&0&r&q\\e&0&s&r\end{vmatrix}}$.

${\begin{aligned}A_{1}&=r{\begin{vmatrix}r&q&p\\s&r&q\\0&s&r\end{vmatrix}}-s{\begin{vmatrix}q&p&0\\s&r&q\\0&s&r\end{vmatrix}}\\&=r(r^{3}+ps^{2}-2qrs)-s(qr^{2}-q^{2}s-prs)\\&=2prs^{2}+q^{2}s^{2}-3qr^{2}s+r^{4}.\end{aligned}}$

${\begin{aligned}A_{2}&=b{\begin{vmatrix}r&q&p\\s&r&q\\0&s&r\end{vmatrix}}-q{\begin{vmatrix}c&q&p\\d&r&q\\e&s&r\end{vmatrix}}+p{\begin{vmatrix}c&r&p\\d&s&q\\e&0&r\end{vmatrix}}\\&=b(r^{3}+ps^{2}-2qrs)-q(cr^{2}+dps+eq^{2}-cqs-dqr-epr)+p(crs+eqr-dr^{2}-eps)\\&=bps^{2}-2bqrs+br^{3}+cprs+cq^{2}s-cqr^{2}-dpqs-dpr^{2}+dq^{2}r-ep^{2}s+2epqr-eq^{3}.\end{aligned}}$

${\begin{aligned}A_{3}&=-r{\begin{vmatrix}c&q&p\\d&r&q\\e&s&r\end{vmatrix}}+s{\begin{vmatrix}b&p&0\\d&r&q\\e&s&r\end{vmatrix}}\\&=-r(cr^{2}+dps+eq^{2}-cqs-dqr-epr)+s(br^{2}+epq-bqs-dpr)\\&=-bqs^{2}+br^{2}s+cqrs-cr^{3}-2dprs+dqr^{2}+epqs+epr^{2}-eq^{2}r.\end{aligned}}$

${\begin{aligned}A&=aA_{1}-qA_{2}+pA_{3}\\&=a(2prs^{2}+q^{2}s^{2}-3qr^{2}s+r^{4})\\&\quad -q(bps^{2}-2bqrs+br^{3}+cprs+cq^{2}s-cqr^{2}-dpqs-dpr^{2}+dq^{2}r-ep^{2}s+2epqr-eq^{3})\\&\quad \quad +p(-bqs^{2}+br^{2}s+cqrs-cr^{3}-2dprs+dqr^{2}+epqs+epr^{2}-eq^{2}r)\\&=2aprs^{2}+aq^{2}s^{2}-3aqr^{2}s+ar^{4}\\&\quad -2bpqs^{2}+bpr^{2}s+2bq^{2}rs-bqr^{3}-cpr^{3}-cq^{3}s+cq^{2}r^{2}\\&\quad \quad -2dp^{2}rs+dpq^{2}s+2dpqr^{2}-dq^{3}r+2ep^{2}qs+ep^{2}r^{2}-3epq^{2}r+eq^{4}.\end{aligned}}$

 

Calcul de $B$

$B=qB_{1}-rB_{2}+sB_{3}$ avec

$B_{1}={\begin{vmatrix}b&a&q&p\\c&b&r&q\\d&c&s&r\\e&d&0&s\end{vmatrix}},\quad B_{2}={\begin{vmatrix}a&0&p&0\\c&b&r&q\\d&c&s&r\\e&d&0&s\end{vmatrix}},\quad B_{3}={\begin{vmatrix}a&0&p&0\\b&a&q&p\\d&c&s&r\\e&d&0&s\end{vmatrix}}$.

${\begin{aligned}B_{1}&=-e{\begin{vmatrix}a&q&p\\b&r&q\\c&s&r\end{vmatrix}}+d{\begin{vmatrix}b&q&p\\c&r&q\\d&s&r\end{vmatrix}}+s{\begin{vmatrix}b&a&q\\c&b&r\\d&c&s\end{vmatrix}}\\&=-e(ar^{2}+bps+cq^{2}-aqs-bqr-cpr)+d(br^{2}+cps+dq^{2}-bqs-cqr-dpr)+s(b^{2}s+c^{2}q+adr-bcr-acs-bdq)\\&=-acs^{2}+adrs+aeqs-aer^{2}+b^{2}s^{2}-bcrs-2bdqs+bdr^{2}-beps+beqr+c^{2}qs+cdps-cdqr+cepr-ceq^{2}-d^{2}pr+d^{2}q^{2}.\end{aligned}}$

${\begin{aligned}B_{2}&=a{\begin{vmatrix}b&r&q\\c&s&r\\d&0&s\end{vmatrix}}+p{\begin{vmatrix}c&b&q\\d&c&r\\e&d&s\end{vmatrix}}\\&=a(bs^{2}+dr^{2}-crs-dqs)+p(c^{2}s+d^{2}q+ber-cdr-bds-ceq)\\&=abs^{2}-acrs-adqs+adr^{2}-bdps+bepr+c^{2}ps-cdpr-cepq+d^{2}pq.\end{aligned}}$

${\begin{aligned}B_{3}&=a{\begin{vmatrix}a&q&p\\c&s&r\\d&0&s\end{vmatrix}}+p{\begin{vmatrix}b&a&p\\d&c&r\\e&d&s\end{vmatrix}}\\&=a(as^{2}+dqr-cqs-dps)+p(bcs+d^{2}p+aer-bdr-ads-cep)\\&=a^{2}s^{2}-acqs-2adps+adqr+aepr+bcps-bdpr-cep^{2}+d^{2}p^{2}.\end{aligned}}$

${\begin{aligned}B&=qB_{1}-rB_{2}+sB_{3}\\&=q(-acs^{2}+adrs+aeqs-aer^{2}+b^{2}s^{2}-bcrs-2bdqs+bdr^{2}-beps+beqr+c^{2}qs+cdps-cdqr+cepr-ceq^{2}-d^{2}pr+d^{2}q^{2})\\&\quad -r(abs^{2}-acrs-adqs+adr^{2}-bdps+bepr+c^{2}ps-cdpr-cepq+d^{2}pq)\\&\quad \quad +s(a^{2}s^{2}-acqs-2adps+adqr+aepr+bcps-bdpr-cep^{2}+d^{2}p^{2})\\&=a^{2}s^{3}-abrs^{2}-2acqs^{2}+acr^{2}s-2adps^{2}+3adqrs-adr^{3}+aeprs+aeq^{2}s-aeqr^{2}\\&\quad +b^{2}qs^{2}+bcps^{2}-bcqrs-2bdq^{2}s+bdqr^{2}-bepqs-bepr^{2}+beq^{2}r\\&\quad \quad -c^{2}prs+c^{2}q^{2}s+cdpqs+cdpr^{2}-cdq^{2}r-cep^{2}s+2cepqr-ceq^{3}+d^{2}p^{2}s-2d^{2}pqr+d^{2}q^{3}.\end{aligned}}$

 

Calcul de $C$

$C=bC_{1}-aC_{2}+pC_{3}$ avec

$C_{1}={\begin{vmatrix}b&q&p&0\\c&r&q&p\\d&s&r&q\\e&0&s&r\end{vmatrix}},\quad C_{2}={\begin{vmatrix}c&q&p&0\\d&r&q&p\\e&s&r&q\\0&0&s&r\end{vmatrix}},\quad C_{3}={\begin{vmatrix}c&b&p&0\\d&c&q&p\\e&d&r&q\\0&e&s&r\end{vmatrix}}$

${\begin{aligned}C_{1}&=-e{\begin{vmatrix}q&p&0\\r&q&p\\s&r&q\end{vmatrix}}-s{\begin{vmatrix}b&q&0\\c&r&p\\d&s&q\\\end{vmatrix}}+r{\begin{vmatrix}b&q&p\\c&r&q\\d&s&r\end{vmatrix}}\\&=-e(q^{3}+p^{2}s-2pqr)-s(bqr+dpq-bps-cq^{2})+r(br^{2}+cps+dq^{2}-bqs-cqr-dpr)\\&=bps^{2}-2bqrs+br^{3}+cprs+cq^{2}s-cqr^{2}-dpqs-dpr^{2}+dq^{2}r-ep^{2}s+2epqr-eq^{3}.\end{aligned}}$

${\begin{aligned}C_{2}&=-s{\begin{vmatrix}c&q&0\\d&r&p\\e&s&q\end{vmatrix}}+r{\begin{vmatrix}c&q&p\\d&r&q\\e&s&r\end{vmatrix}}\\&=-s(cqr+epq-cps-dq^{2})+r(cr^{2}+dps+eq^{2}-cqs-dqr-epr)\\&=cps^{2}-2cqrs+cr^{3}+dprs+dq^{2}s-dqr^{2}-epqs-epr^{2}+eq^{2}r.\end{aligned}}$

${\begin{aligned}C_{3}&=e{\begin{vmatrix}c&p&0\\d&q&p\\e&r&q\end{vmatrix}}-s{\begin{vmatrix}c&b&0\\d&c&p\\e&d&q\end{vmatrix}}+r{\begin{vmatrix}c&b&p\\d&c&q\\e&d&r\end{vmatrix}}\\&=e(cq^{2}+ep^{2}-cpr-dpq)-s(c^{2}q+bep-cdp-bdq)+r(c^{2}r+d^{2}p+beq-cdq-bdr-cep)\\&=bdqs-bdr^{2}-beps+beqr-c^{2}qs+c^{2}r^{2}+cdps-cdqr-2cepr+ceq^{2}+d^{2}pr-depq+e^{2}p^{2}.\end{aligned}}$

${\begin{aligned}C&=bC_{1}-aC_{2}+pC_{3}\\&=b(bps^{2}-2bqrs+br^{3}+cprs+cq^{2}s-cqr^{2}-dpqs-dpr^{2}+dq^{2}r-ep^{2}s+2epqr-eq^{3})\\&\quad -a(cps^{2}-2cqrs+cr^{3}+dprs+dq^{2}s-dqr^{2}-epqs-epr^{2}+eq^{2}r)\\&\quad \quad +p(bdqs-bdr^{2}-beps+beqr-c^{2}qs+c^{2}r^{2}+cdps-cdqr-2cepr+ceq^{2}+d^{2}pr-depq+e^{2}p^{2})\\&=-acps^{2}+2acqrs-acr^{3}-adprs-adq^{2}s+adqr^{2}+aepqs+aepr^{2}-aeq^{2}r\\&\quad +b^{2}ps^{2}-2b^{2}qrs+b^{2}r^{3}+bcprs+bcq^{2}s-bcqr^{2}-2bdpr^{2}+bdq^{2}r-2bep^{2}s+3bepqr-beq^{3}\\&\quad \quad -c^{2}pqs+c^{2}pr^{2}+cdp^{2}s-cdpqr-2cep^{2}r+cepq^{2}+d^{2}p^{2}r-dep^{2}q+e^{2}p^{3}.\end{aligned}}$

 

Calcul de $D$

$D=bD_{1}-aD_{2}-pD_{3}$ avec

$D_{1}={\begin{vmatrix}b&a&q&p\\c&b&r&q\\d&c&s&r\\e&d&0&s\end{vmatrix}},\quad D_{2}={\begin{vmatrix}c&a&q&p\\d&b&r&q\\e&c&s&r\\0&d&0&s\end{vmatrix}},\quad D_{3}={\begin{vmatrix}c&b&a&p\\d&c&b&q\\e&d&c&r\\0&e&d&s\end{vmatrix}}$.

${\begin{aligned}D_{1}&=-e{\begin{vmatrix}a&q&p\\b&r&q\\c&s&r\end{vmatrix}}+d{\begin{vmatrix}b&q&p\\c&r&q\\d&s&r\end{vmatrix}}+s{\begin{vmatrix}b&a&q\\c&b&r\\d&c&s\end{vmatrix}}\\&=-e(ar^{2}+bps+cq^{2}-aqs-bqr-cpr)+d(br^{2}+cps+dq^{2}-bqs-cqr-dpr)+s(b^{2}s+c^{2}q+adr-bcr-acs-bdq)\\&=-acs^{2}+adrs+aeqs-aer^{2}+b^{2}s^{2}-bcrs-2bdqs+bdr^{2}-beps+beqr+c^{2}qs+cdps-cdqr+cepr-ceq^{2}-d^{2}pr+d^{2}q^{2}.\end{aligned}}$

${\begin{aligned}D_{2}&=d{\begin{vmatrix}c&q&p\\d&r&q\\e&s&r\end{vmatrix}}+s{\begin{vmatrix}c&a&q\\d&b&r\\e&c&s\end{vmatrix}}\\&=d(cr^{2}+dps+eq^{2}-cqs-dqr-epr)+s(bcs+cdq+aer-c^{2}r-ads-beq)\\&=-ads^{2}+aers+bcs^{2}-beqs-c^{2}rs+cdr^{2}+d^{2}ps-d^{2}qr-depr+deq^{2}.\end{aligned}}$

${\begin{aligned}D_{3}&=e{\begin{vmatrix}c&a&p\\d&b&q\\e&c&r\end{vmatrix}}-d{\begin{vmatrix}c&b&p\\d&c&q\\e&d&r\end{vmatrix}}+s{\begin{vmatrix}c&b&a\\d&c&b\\e&d&c\end{vmatrix}}\\&=e(bcr+cdp+aeq-c^{2}q-adr-bep)-d(c^{2}r+d^{2}p+beq-cdq-bdr-cep)+s(c^{3}+ad^{2}+b^{2}e-2bcd-ace)\\&=-aces+ad^{2}s-ader+ae^{2}q+b^{2}es-2bcds+bcer+bd^{2}r-bdeq-be^{2}p+c^{3}s-c^{2}dr-c^{2}eq+cd^{2}q+2cdep-d^{3}p.\end{aligned}}$

${\begin{aligned}D&=bD_{1}-aD_{2}-pD_{3}\\&=b(-acs^{2}+adrs+aeqs-aer^{2}+b^{2}s^{2}-bcrs-2bdqs+bdr^{2}-beps+beqr+c^{2}qs+cdps-cdqr+cepr-ceq^{2}-d^{2}pr+d^{2}q^{2})\\&\quad -a(-ads^{2}+aers+bcs^{2}-beqs-c^{2}rs+cdr^{2}+d^{2}ps-d^{2}qr-depr+deq^{2})\\&\quad \quad -p(-aces+ad^{2}s-ader+ae^{2}q+b^{2}es-2bcds+bcer+bd^{2}r-bdeq-be^{2}p+c^{3}s-c^{2}dr-c^{2}eq+cd^{2}q+2cdep-d^{3}p)\\&=a^{2}ds^{2}-a^{2}ers-2abcs^{2}+abdrs+2abeqs-aber^{2}+ac^{2}rs-acdr^{2}+aceps-2ad^{2}ps+ad^{2}qr+2adepr-adeq^{2}-ae^{2}pq\\&\quad +b^{3}s^{2}-b^{2}crs-2b^{2}dqs+b^{2}dr^{2}-2b^{2}eps+b^{2}eqr+bc^{2}qs+3bcdps-bcdqr-bceq^{2}-2bd^{2}pr+bd^{2}q^{2}+bdepq+be^{2}p^{2}\\&\quad \quad -c^{3}ps+c^{2}dpr+c^{2}epq-cd^{2}pq-2cdep^{2}+d^{3}p^{2}.\end{aligned}}$

 

${\begin{aligned}\operatorname {Res} (P,Q)&=ae\left[{\begin{aligned}&2aprs^{2}+aq^{2}s^{2}-3aqr^{2}s+ar^{4}\\&-2bpqs^{2}+bpr^{2}s+2bq^{2}rs-bqr^{3}-cpr^{3}-cq^{3}s+cq^{2}r^{2}\\&-2dp^{2}rs+dpq^{2}s+2dpqr^{2}-dq^{3}r+2ep^{2}qs+ep^{2}r^{2}-3epq^{2}r+eq^{4}\end{aligned}}\right]\\&+as\left[{\begin{aligned}&a^{2}s^{3}-abrs^{2}-2acqs^{2}+acr^{2}s-2adps^{2}+3adqrs-adr^{3}+aeprs+aeq^{2}s-aeqr^{2}\\&+b^{2}qs^{2}+bcps^{2}-bcqrs-2bdq^{2}s+bdqr^{2}-bepqs-bepr^{2}+beq^{2}r\\&-c^{2}prs+c^{2}q^{2}s+cdpqs+cdpr^{2}-cdq^{2}r-cep^{2}s+2cepqr-ceq^{3}+d^{2}p^{2}s-2d^{2}pqr+d^{2}q^{3}\end{aligned}}\right]\\&+pe\left[{\begin{aligned}&-acps^{2}+2acqrs-acr^{3}-adprs-adq^{2}s+adqr^{2}+aepqs+aepr^{2}-aeq^{2}r\\&+b^{2}ps^{2}-2b^{2}qrs+b^{2}r^{3}+bcprs+bcq^{2}s-bcqr^{2}-2bdpr^{2}+bdq^{2}r-2bep^{2}s+3bepqr-beq^{3}\\&-c^{2}pqs+c^{2}pr^{2}+cdp^{2}s-cdpqr-2cep^{2}r+cepq^{2}+d^{2}p^{2}r-dep^{2}q+e^{2}p^{3}\end{aligned}}\right]\\&-ps\left[{\begin{aligned}&a^{2}ds^{2}-a^{2}ers-2abcs^{2}+abdrs+2abeqs-aber^{2}+ac^{2}rs-acdr^{2}+aceps-2ad^{2}ps+ad^{2}qr+2adepr-adeq^{2}-ae^{2}pq\\&+b^{3}s^{2}-b^{2}crs-2b^{2}dqs+b^{2}dr^{2}-2b^{2}eps+b^{2}eqr+bc^{2}qs+3bcdps-bcdqr-bceq^{2}-2bd^{2}pr+bd^{2}q^{2}+bdepq+be^{2}p^{2}\\&-c^{3}ps+c^{2}dpr+c^{2}epq-cd^{2}pq-2cdep^{2}+d^{3}p^{2}\end{aligned}}\right]\\&\\&=a^{2}(as^{4}-brs^{3}-2cqs^{3}+cr^{2}s^{2}-3dps^{3}+3dqrs^{2}-dr^{3}s+4eprs^{2}+2eq^{2}s^{2}-4eqr^{2}s+er^{4})\\&+ab(bqs^{3}+3cps^{3}-cqrs^{2}-dprs^{2}-2dq^{2}s^{2}+dqr^{2}s-5epqs^{2}+epr^{2}s+3eq^{2}rs-eqr^{3})\\&+ac(-2cprs^{2}+cq^{2}s^{2}+dpqs^{2}+2dpr^{2}s-dq^{2}rs-3ep^{2}s^{2}+4epqrs-2epr^{3}-2eq^{3}s+eq^{2}r^{2})\\&+ad(3dp^{2}s^{2}-3dpqrs+dq^{3}s-5ep^{2}rs+epq^{2}s+3epqr^{2}-eq^{3}r)\\&+ae^{2}(4p^{2}qs+2p^{2}r^{2}-4pq^{2}r+q^{4})\\&+bp\left[b(-bs^{3}+crs^{2}+2dqs^{2}-dr^{2}s+3eps^{2}-3eqrs+er^{3})+c(-cqs^{2}-3dps^{2}+dqrs+eprs+2eq^{2}s-eqr^{2})\right]\\&+bp\left[d(2dprs-dq^{2}s-epqs-2epr^{2}+eq^{2}r)+e^{2}(-3p^{2}s+3pqr-q^{3})\right]\\&+cp^{2}\left[c(cs^{2}-drs-2eqs+er^{2})+d(dqs+3eps-eqr)+e^{2}(-2pr+q^{2})\right]\\&+dp^{3}(-d^{2}s+der-e^{2}q)+e^{3}p^{4}.\end{aligned}}$

On trouve le même résultat que dans l'exercice 1-1.