% % macros for differentials, partial derivatives, Jacobian matrices \def\vectorsmal#1{#1} \def\vecsmal#1{\vbox{\ialign{##\crcr \rightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip} $\hfil\scriptstyle{#1}\hfil$\crcr}}} \def\monom#1#2#3{#1_{1}^{#2_{1}}\cdots #1_{#3}^{#2_{#3}}} \def\Sym#1{\hbox{Sym}^#1} \def\Ispac#1#2{#1^{(#2)}} %\def\affspac#1#2{{\bf A}^{#1}_{#2}} \def\affspac#1#2{{\mathbb A}^{#1}_{#2}} %\def\affreal{{\bf A}} \def\affreal{{\mathbb A}} \def\fixset{\hbox{Fix}} \def\mod{\hbox{mod}} \def\sgnam{\hbox{sg}} \def\volum{\hbox{vol}} \def\Lambatp#1{\Lambda_{\mathrm{at}\>#1}} \def\Rat#1#2{\mathrm{Rat}(#1,#2)} \def\Mer#1{\s{M}\mathrm{er}(#1)} \def\Irred#1{\mathrm{Irred}/#1} \def\FncFlds#1{\mathrm{FncFlds}/#1} \def\trace{\mathrm{tr}} \def\Sing{\mathrm{Sing}} \def\qtrace{\hbox{Tr}} \def\diag{\hbox{diag}} \def\Gram{\hbox{Gram}} \def\Ad{\mathrm{Ad}} \def\Frac{\mathrm{Frac}} \def\app{\hbox{app}} \def\smad{\mathrm{ad}} %\def\eucreal{{\bf E}} \def\eucreal{{\mathbb E}} \def\qone{{\bf 1}} \def\qi{{\bf i}} \def\qj{{\bf j}} \def\qk{{\bf k}} \def\quat{{\mathbb H}} \def\setscat{\mathbf{Sets}} \def\grcat{\mathbf{Grp}} \def\abcat{\mathbf{Ab}} \def\crngcat{\mathbf{CRng}} \def\kalgcat{\hbox{\bf k-alg}} \def\affalgcat{\hbox{\bf k-affalg}} \def\Ob{\mathrm{Ob}} \def\Mor{\mathrm{Mor}} \def\Ar{\mathrm{Ar}} % \def\res{|} \def\brokrarrow{\>\raise1.5pt\hbox{$\scriptstyle -\, -\, \rightarrow$}\>} \def\ratmap#1#2#3{#1\co #2 \brokrarrow #3} %\def\ratmap#1#2#3{#1\co #2 - -\rightarrow #3} \def\blowup#1#2{\hbox{Bl}_{#1} #2} \def\mGL#1{{\bf GL}(#1)} \def\mGA#1{{\bf GA}(#1)} \def\mGAa#1{{\bf GA}_a(#1)} \def\cech#1{\check{#1}} \def\mE#1{{\bf E}(#1)} \def\mIs#1{{\bf Is}(#1)} \def\mMo#1{{\bf Mo}(#1)} \def\mPGL#1{{\bf PGL}(#1)} \def\mSL#1{{\bf SL}(#1)} \def\mEA#1{{\bf EA}(#1)} \def\mIA#1{{\bf IA}(#1)} \def\mSE#1{{\bf SE}(#1)} \def\mU#1{{\bf U}(#1)} \def\mUpq#1#2{{\bf U}(#1,#2)} \def\mSU#1{{\bf SU}(#1)} \def\mSUpq#1#2{{\bf SU}(#1,#2)} \def\mSA#1{{\bf SA}(#1)} \def\mPSL#1{{\bf PSL}(#1)} \def\mO#1{{\bf O}(#1)} \def\mOpq#1#2{{\bf O}(#1,#2)} \def\mSO#1{{\bf SO}(#1)} \def\mSOpq#1#2{{\bf SO}(#1,#2)} \def\mPSO#1{{\bf PSO}(#1)} \def\mAut#1{{\bf Aut}(#1)} \def\mDIL#1{{\bf DIL}(#1)} \def\Grassm{{\bf G}} \def\PGrassm{{\bf PG}} \def\zozo#1{\vector{#1}} \def\zozor#1{\novect{#1}} \def\affvec#1{\vector{#1}} \def\intd{\int\!\!\!\int} \def\hli#1{\widehat{#1}} \def\osc#1#2{{\rm Osc}_{#1} #2} \def\Span{{\rm Span\/}} \def\fallpow#1#2{#1^{\underline{#2}}} \def\ringpoly#1#2{#1[#2]} \def\fracfpoly#1#2{#1(#2)} \def\ftranspos#1{\hbox{$^{t}#1$}} \def\transpos#1{#1^{\top}} \def\flecheabov#1{\buildrel #1\over\longrightarrow} \def\flechea#1#2#3{#1\,\flecheabov{#2}\,#3} \def\flecheb#1#2#3#4#5{#1\,\flecheabov{#2}\,#3\,\flecheabov{#4}\,#5} \def\flechec#1#2#3#4#5#6#7{#1\,\flecheabov{#2}\,#3 \,\flecheabov{#4}\,#5\,\flecheabov{#6}\,#7} \def\fleched#1#2#3#4#5#6#7#8#9{#1\,\flecheabov{#2}\,#3\,\flecheabov{#4}\,#5 \,\flecheabov{#6}\,#7\,\flecheabov{#8}\,#9} \def\flechel#1#2{#1\,\flecheabov{#2}\,} \def\flecher#1#2{\,\flecheabov{#1}\,#2} \def\shortexact#1#2#3#4#5{0\,\flecheabov{}\,#1\,\flecheabov{#2}\,#3 \,\flecheabov{#4}\,#5\,\flecheabov{}\,0} \def\libvec#1#2{\overrightarrow{#1#2}} \def\toto#1#2{\overrightarrow{#1#2}} \def\libvecb#1#2{#2 - #1} \def\libvecbo#1#2{{\bf #1#2}} %\def\vectorr#1{\overrightarrow{#1}} \def\vectorr#1{#1} \def\bvect#1{{\bf #1}} \def\novect#1{#1} \def\vector#1{\fakoverrightarrow{#1}} \def\ptb#1{\overline{#1}} \def\covec#1{#1^{*}} \def\ortho#1{#1^{0}} \def\orthog#1{#1^{\perp}} \def\biortho#1{#1^{00}} \def\biorthog#1{#1^{\perp\perp}} \def\translat#1#2{#1 + \novect{#2}} \def\translatv#1#2{#1 + \vector{#2}} \def\binvec#1#2{\pmatrix{#1\cr #2\cr}} \def\trivec#1#2#3{\pmatrix{#1\cr #2\cr #3\cr}} \def\bincoef#1#2{#1\choose #2} \def\mapdef#1#2#3{#1\co #2\rightarrow #3} \def\famil#1#2#3{(#1_{#2})_{#2\in #3}} \def\familvec#1#2#3{(\vector{#1_{#2}})_{#2\in #3}} \def\linspac#1#2{{\rm L}(#1; #2)} \def\slinspac#1#2{{\rm S}(#1; #2)} \def\clinspac#1#2#3{{\cal L}_{#1}(#2; #3)} \def\lincomb#1#2#3#4{\sum_{#3\in #4} #1_{#3}\vector{#2_{#3}}} \def\lincombin#1#2#3#4{\sum_{#3\in #4} #1_{#3} #2_{#3}} \def\linsum#1#2#3{#1_{1}\vector{#2_{1}} + \cdots + #1_{#3}\vector{#2_{#3}}} \def\linsom#1#2#3{#1_{1}#2_{1} + \cdots + #1_{#3}#2_{#3}} \def\pmatrice#1{\left( \matrice{#1} \right)} \def\dmatrice#1{\left| \matrix{#1} \right|} \def\dmatriceb#1{\left| \matrice{#1} \right|} \def\colvec#1#2{\pmatrix{#1_{1}\cr \vdots\cr #1_{#2}\cr}} \def\trivec#1#2#3{\pmatrix{#1\cr #2\cr #3\cr}} \def\colvector#1#2{\pmatrix{#1_{1}\cr #1_{2}\cr\vdots\cr #1_{#2}\cr}} \def\linvec#1#2{(#1_{1},\ldots, #1_{#2})} \def\specrow#1#2#3{(\matel{#1}{#2}{1},\ldots,\matel{#1}{#2}{#3})} \def\speccol#1#2#3{\pmatrix{\matel{#1}{1}{#2}\cr \vdots\cr \matel{#1}{#3}{#2}\cr}} \def\linbasis#1#2{(\vector{#1_{1}},\ldots, \vector{#1_{#2}})} \def\lbasis#1#2{(#1_{1},\ldots, #1_{#2})} \def\eunorme#1#2{\left(|#1_{1}|^{2} + \cdots + |#1_{#2}|^{2}\right)^{1\over 2}} \def\eunorm#1#2{\sqrt{|#1_{1}|^{2} + \cdots + |#1_{#2}|^{2})}} \def\eudist#1#2#3{\left(|#1_{1} - #2_{1}|^{2}+ \cdots + |#1_{#3} - #2_{#3}|^{2}\right)^{1\over 2}} \def\norme#1{\left\|#1\right\|} \def\smnorme#1{\|#1\|} \def\dist#1#2#3{d_{#1}(#2,\,#3)} \def\inprod#1#2{(#1 | #2)} \def\dotprod#1#2{#1\cdot #2} \def\absval#1#2{|#1 - #2|} \def\cloball#1#2{B(#1,#2)} \def\opball#1#2{B_{0}(#1,#2)} \def\ncloball#1{B(#1)} \def\nopball#1{B_{0}(#1)} \def\adher#1{\overline{#1}} \def\interio#1{\buildrel \circ\over #1} \def\fr#1{{\rm Fr}\>#1} % \def\polynom#1#2#3{#1_{0}+#1_{1}#3+\cdots +#1_{#2}#3^{#2}} \def\hpolynom#1#2#3{#1_{#2}#3^{#2} + #1_{#2 - 1}#3^{#2 - 1} + \cdots + #1_{0}} \def\derivpol#1#2#3{#2#1_{#2}#3^{#2 - 1} + (#2 - 1)#1_{#2 - 1}#3^{#2 - 2} + \cdots + 2#1_{2}#3 + #1_{1}} \def\rpolynom#1#2#3{#1_{0}#3^{#2} + #1_{1}#3^{#2 - 1} + \cdots + #1_{#2}} \def\monic#1#2#3{#3^{#2} + #1_{#2 - 1}#3^{#2 - 1} + \cdots + #1_{0}} \def\rmonic#1#2#3{#3^{#2} + #1_{1}#3^{#2 - 1} + \cdots + #1_{#2}} \def\pcoef#1#2#3{#1_{(#2_{1},\ldots,#2_{#3})}} \def\mpolynom#1#2#3#4{\sum_{(#2_{1},\ldots,#2_{#3})\in\natnums^{(#3)}} \pcoef{#1}{#2}{#3}\monom{#4}{#2}{#3}} \def\myfrac#1#2{{\displaystyle {#1\strut\over\strut #2}}} %\def\dfrac#1#2{{\displaystyle #1 \strut\over\strut \displaystyle #2}} \def\dfrac#1#2{{\displaystyle {{\displaystyle #1}\strut\over\strut #2}}} \def\batop#1#2{{\displaystyle #1\atop \displaystyle #2}} \def\derivpderivDo#1{#1'} \def\derivp#1{#1'} \def\derivD#1#2{{\rm D}^{#2}#1} \def\derivDof#1#2{{\rm D}#1#2} \def\vfderiv#1#2{{\rm D}_{#1}#2} \def\parderiv#1#2{\displaystyle{{\partial{#1}}\strut\over\strut{\partial{#2}}}} \def\parderivb#1#2{\displaystyle{{\partial^2{#1}} \over\strut{\partial{#2^{2}}}}} \def\parderivc#1#2#3{\displaystyle{{\partial^2{#1}} \strut\over\strut{\partial{#2}\partial{#3}}}} \def\parderivd#1#2#3#4#5{\displaystyle{{\partial^{#2}{#1}} \strut\over\strut{\partial{#3_{#4}}\ldots\partial{#3_{#5}}}}} \def\vparderiv#1#2{\displaystyle{{\partial{#1}} \strut\over\strut{\partial{\vector{#2}}}}} % \def\vparderivh#1#2#3{\displaystyle{{\partial^{#3} {#1}} \strut\over\strut{\partial{\vector{#2}}^{#3}}}} % % \def\vparderivb#1#2#3{\displaystyle{% \partial^{2} {#1}\strut\over\strut{% \partial{\vector{#2}}\partial{\vector{#3}}% }% }% } % \def\vparderivc#1#2#3#4#5{\displaystyle{% \partial^{#4 + #5} {#1}\strut\over\strut{% \partial{\vector{#2}}^{#4}\partial{\vector{#3}}^{#5}% }% }% } % \def\Der#1#2#3#4{\mathrm{Der}_{#1}(#2, #3; #4)} \def\sDer#1#2#3{\mathrm{Der}_{#1}(#2, #3)} \def\Dparderiv#1#2{\partial_{#2}{#1}} \def\Dparder#1#2{{\rm D}_{#2}{#1}} \def\deriv#1#2{{\displaystyle{d#1\strut\over\strut d#2}}} \def\dotderiv#1{{\dot #1}} \def\dotdderiv#1{{\ddot #1}} \def\vderiv#1#2{{\rm D}_{\vectorsmal{#1}}{#2}} % \def\vderivb#1#2#3{{\rm D}^{2}_{\vectorsmal{#1},\vectorsmal{#2}}{#3}} \def\vderivc#1#2#3{{\rm D}_{\vectorsmal{#1}}{\rm D}_{\vectorsmal{#2}}{#3}} \def\vderivd#1#2#3{{\rm D}_{\vectorsmal{#1}}\ldots {\rm D}_{\vectorsmal{#2}}{#3}} % \def\vderivdd#1#2#3#4#5{\underbrace{{\rm D}_{\vectorsmal{#1}}\ldots {\rm D}_{\vectorsmal{#2}}}_{i}\underbrace{{\rm D}_{\vectorsmal{#3}}\ldots {\rm D}_{\vectorsmal{#4}}}_{j}\,{#5}} % \def\matdef#1#2{{\rm M}_{#1}#2} \def\matdefin#1#2#3{{\rm M}_{#1, #2}#3} % \def\jacob#1#2#3#4#5{% \pmatrix{\parderiv{#1_{1}}{#2_{1}}(#3)&\parderiv{#1_{1}}{#2_{2}}(#3)&\ldots& \parderiv{#1_{1}}{#2_{#5}}(#3)\cr \parderiv{#1_{2}}{#2_{1}}(#3)&\parderiv{#1_{2}}{#2_{2}}(#3)&\ldots& \parderiv{#1_{2}}{#2_{#5}}(#3)\cr \vdots&\vdots&\ddots&\vdots\cr \parderiv{#1_{#4}}{#2_{1}}(#3)&\parderiv{#1_{#4}}{#2_{2}}(#3)&\ldots& \parderiv{#1_{#4}}{#2_{#5}}(#3)\cr}} % % \def\Djacob#1#2#3#4{% \pmatrix{\Dparderiv{#1_{1}}{1}(#4)&\Dparderiv{#1_{1}}{2}(#4)&\ldots& \Dparderiv{#1_{1}}{#3}(#4)\cr \Dparderiv{#1_{2}}{1}(#4)&\Dparderiv{#1_{2}}{2}(#4)&\ldots& \Dparderiv{#1_{2}}{#3}(#4)\cr \vdots&\vdots&\ddots&\vdots\cr \Dparderiv{#1_{#2}}{1}(#4)&\Dparderiv{#1_{#2}}{2}(#4)&\ldots& \Dparderiv{#1_{#2}}{#3}(#4)\cr}} % % \def\cjacob#1#2#3#4#5{% \pmatrix{\parderiv{#1}{#4}(#5)\cr \parderiv{#2}{#4}(#5)\cr \parderiv{#3}{#4}(#5)\cr}} % % \def\ljacob#1#2#3#4#5{% \bigg(\parderiv{#1}{#2}(#5)\>\> \parderiv{#1}{#3}(#5)\>\> \parderiv{#1}{#4}(#5)\bigg)} % % \def\jacoba#1#2#3#4#5{% \pmatrix{\parderiv{#1}{#3}(#5)&\parderiv{#1}{#4}(#5)\cr \parderiv{#2}{#3}(#5)&\parderiv{#2}{#4}(#5)\cr}} % % \def\jacobb#1#2#3#4#5#6{% \pmatrix{\parderiv{#1}{#4}(#6)&\parderiv{#1}{#5}(#6)\cr \parderiv{#2}{#4}(#6)&\parderiv{#2}{#5}(#6)\cr \parderiv{#3}{#4}(#6)&\parderiv{#3}{#5}(#6)\cr}} % \def\rowjacob#1#2#3#4{% \bigg(\parderiv{#1}{#2_{1}}(#3),\ldots, \parderiv{#1}{#2_{#4}}(#3)\bigg)} % \def\hessien#1#2#3#4{% \pmatrix{\parderivb{#1}{#2_{1}}(#3)&\parderivc{#1}{#2_{1}}{#2_{2}}(#3)&\ldots& \parderivc{#1}{#2_{1}}{#2_{#4}}(#3)\cr \parderivc{#1}{#2_{1}}{#2_{2}}(#3)&\parderivb{#1}{#2_{2}}(#3)&\ldots& \parderivc{#1}{#2_{2}}{#2_{#4}}(#3)\cr \vdots&\vdots&\ddots&\vdots\cr \parderivc{#1}{#2_{1}}{#2_{#4}}(#3)&\parderivc{#1}{#2_{2}}{#2_{#4}}(#3)&\ldots& \parderivb{#1}{#2_{#4}}(#3)\cr}} % % \def\matel#1#2#3{#1_{#2\, #3}} % %\def\mata#1#2#3#4{% %\pmatrice{#1& #2\cr %#3\cr}} \def\mata#1#2#3#4{% \pmatrice{#1& #2\cr #3\cr}} % \def\matta#1#2#3#4{% \left( \matrix{ #1 & #2\cr #3 & #4\cr } \right)} % % \def\matb#1{% \pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}& \matel{#1}{1}{3}\cr \matel{#1}{2}{1}& \matel{#1}{2}{2}& \matel{#1}{2}{3}\cr \matel{#1}{3}{1}& \matel{#1}{3}{2}& \matel{#1}{3}{3}\cr}} % % \def\matc#1#2#3#4#5#6#7#8#9{% \pmatrice{#1& #2& #3\cr #4 & #5 & #6\cr #7 & #8 & #9\cr}} % \def\mattc#1#2#3#4#5#6#7#8#9{% \left( \matrix{ #1 & #2 & #3\cr #4 & #5 & #6\cr #7 & #8 & #9\cr } \right)} % % \def\genmat#1#2#3{% \pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#3}\cr \matel{#1}{2}{1}& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#3}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#3}\cr}} % \def\genmatrix#1#2#3{% \pmatrix{\matel{#1}{1}{1}& \ldots& \matel{#1}{1}{#3}\cr \vdots&\ddots&\vdots\cr \matel{#1}{#2}{1}& \ldots& \matel{#1}{#2}{#3}\cr}} % \def\idmat{% \pmatrix{\matel{1}{}{}& \matel{0}{}{}&\ldots& \matel{0}{}{}\cr \matel{0}{}{}& \matel{1}{}{}&\ldots& \matel{0}{}{}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{0}{}{}& \matel{0}{}{}&\ldots& \matel{1}{}{}\cr}} % % \def\uptmat#1#2#3{% \pmatrix{ \matel{#1}{1}{1}&\matel{#1}{1}{2}&\matel{#1}{1}{3}&\ldots& \matel{#1}{1}{#3 -1}&\matel{#1}{1}{#3}\cr 0& \matel{#1}{2}{2}&\matel{#1}{2}{3}&\ldots& \matel{#1}{2}{#3 -1}& \matel{#1}{2}{#3}\cr 0& 0&\matel{#1}{3}{3}&\ldots& \matel{#1}{3}{#3 - 1}& \matel{#1}{3}{#3}\cr \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr 0& 0&0&\ldots&\matel{#1}{#2 -1}{#3 -1} &\matel{#1}{#2 - 1}{#3}\cr 0& 0&0&\ldots& 0 &\matel{#1}{#2}{#3}\cr}} % % \def\uptmatb#1#2#3#4{% \pmatrix{ #4_{1} - #4&\matel{#1}{1}{2}&\matel{#1}{1}{3}&\ldots& \matel{#1}{1}{#3 -1}&\matel{#1}{1}{#3}\cr 0& #4_{2} - #4&\matel{#1}{2}{3}&\ldots& \matel{#1}{2}{#3 -1}& \matel{#1}{2}{#3}\cr 0& 0_{3} - #4&\ldots& \matel{#1}{3}{#3 - 1}& \matel{#1}{3}{#3}\cr \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\cr 0& 0&0&\ldots_{#2 - 1} - #4&\matel{#1}{#2 - 1}{#3}\cr 0& 0&0&\ldots& 0 _{#3}- #4\cr}} % % \def\onecolmat#1#2#3{% \pmatrix{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#3}\cr 0& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#3}\cr \vdots&\vdots&\ddots&\vdots\cr 0& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#3}\cr}} % \def\idmatrix{% \pmatrix{\matel{1}{}{}& \matel{0}{}{}&\ldots& \matel{0}{}{}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{0}{}{}& \matel{0}{}{}&\ldots& \matel{1}{}{}\cr}} % \def\diagmat#1#2{% \pmatrix{#1_{1}& &\ldots& \cr & #1_{2} &\ldots& \cr \vdots&\vdots&\ddots&\vdots\cr & &\ldots& #1_{#2}\cr}} % % \def\diagmatso#1#2#3#4{% \pmatrix{ I_{#3}& & & \ldots & \cr & -I_{#4} & & & \cr & & #1_{1} & \ldots & \cr \vdots& &\vdots & \ddots & \vdots \cr & & & \ldots & #1_{#2} \cr} } % % \def\diagmatsob#1#2#3#4{% \pmatrix{ #1_{1} & \ldots & & & \cr \vdots & \ddots & \vdots & &\vdots \cr & \ldots & #1_{#2}& & \cr & & &-I_{#4} & \cr \ldots & & & & I_{#3} \cr } } % % \def\diagmatsobb#1#2#3{% \pmatrix{ #1_{1} & \ldots & & \cr \vdots & \ddots & \vdots & \cr & \ldots & #1_{#2}& \cr \ldots & & & I_{#3}\cr } } % % \def\diagmatlsobb#1#2#3{% \pmatrix{ #1_{1} & \ldots & & \cr \vdots & \ddots & \vdots & \cr & \ldots & #1_{#2}& \cr \ldots & & & 0_{#3}\cr } } % \def\diagmatrecm#1#2{% \pmatrix{#1_{1}& &\ldots& \cr & #1_{2} &\ldots& \cr \vdots&\vdots&\ddots&\vdots\cr & &\ldots& #1_{#2}\cr 0 & \vdots & \ldots & 0\cr \vdots&\vdots&\ddots&\vdots\cr 0 & \vdots & \ldots & 0\cr }} % \def\diagmatrecn#1#2{% \pmatrix{#1_{1}& &\ldots& & 0 &\ldots &0\cr & #1_{2} &\ldots& & 0&\ldots& 0 \cr \vdots&\vdots&\ddots&\vdots& 0 &\vdots & 0\cr & &\ldots& #1_{#2}& 0 & \ldots & 0\cr}} % % \def\kappamatrix{% \pmatrix{ 0 & \kappa_1& & & \cr -\kappa_1& 0 &\kappa_2& & \cr & -\kappa_2& 0 &\ddots & \cr & &\ddots &\ddots &\kappa_{n-1} \cr & & &-\kappa_{n-1} & 0 \cr}} % \def\gendet#1#2{% \dmatrice{\matel{#1}{1}{1}& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#2}\cr \matel{#1}{2}{1}& \matel{#1}{2}{2}&\ldots& \matel{#1}{2}{#2}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}\cr}} % % \def\charpoly#1#2#3{% \dmatrice{\matel{#1}{1}{1} - #3& \matel{#1}{1}{2}&\ldots& \matel{#1}{1}{#2}\cr \matel{#1}{2}{1}& \matel{#1}{2}{2} - #3&\ldots& \matel{#1}{2}{#2}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{#1}{#2}{1}& \matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}- #3\cr}} % % \def\vecdet#1#2#3{% \dmatrice{\matel{#1}{1}{1}& \ldots& \matel{#1}{1}{#2 -1}& #3_{1}\cr \matel{#1}{2}{1}& \ldots&\matel{#1}{2}{#2 -1}& #3_{2}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{#1}{#2}{1}&\ldots& \matel{#1}{#2}{#2 -1}& #3_{#2}\cr}} % % \def\detb#1{% \dmatrice{\matel{#1}{1}{1}& \matel{#1}{1}{2}& \matel{#1}{1}{3}\cr \matel{#1}{2}{1}& \matel{#1}{2}{2}& \matel{#1}{2}{3}\cr \matel{#1}{3}{1}& \matel{#1}{3}{2}& \matel{#1}{3}{3}\cr}} % % \def\detc#1#2#3#4#5#6#7#8#9{% \dmatrice{#1& #2& #3\cr #4& #5& #6\cr #7& #8& #9\cr}} % % \def\detcb#1#2#3#4#5#6#7#8#9{% \dmatriceb{#1& #2& #3\cr #4& #5& #6\cr #7& #8& #9\cr}} % % % \def\deta#1#2#3#4{% \dmatrice{#1& #2\cr #3& #4\cr}} \def\resultant#1#2#3#4{% \dmatrice{ #1_0 & #1_1 & \cdots & \cdots & #1_{#3} & 0 &\cdots &\cdots &\cdots &\cdots & 0\cr 0 & #1_0 & #1_1 & \cdots & \cdots & #1_{#3} & 0 &\cdots &\cdots &\cdots & 0\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr 0 & \cdots &\cdots &\cdots &\cdots & 0 & #1_0 & #1_1 & \cdots & \cdots & #1_{#3}\cr #2_0 & #2_1 & \cdots & \cdots &\cdots &\cdots &\cdots & #2_{#4} & 0 & \cdots & 0\cr 0 & #2_0 & #2_1 & \cdots & \cdots & \cdots &\cdots &\cdots & #2_{#4} & 0 &\cdots\cr \cdots &\cdots &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr 0 & \cdots & 0 & #2_0 & #2_1 & \cdots & \cdots&\cdots &\cdots &\cdots & #2_{#4}\cr }} \def\resultantht#1#2#3#4#5{% \dmatrice{ #1_0 & #5#1_1 & \cdots & \cdots & #5^{#3}#1_{#3} & 0 &\cdots &\cdots &\cdots &\cdots & 0\cr 0 & #1_0 & #5#1_1 & \cdots & \cdots & #5^{#3}#1_{#3} & 0 &\cdots &\cdots &\cdots & 0\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr 0 & \cdots &\cdots &\cdots &\cdots & 0 & #1_0 & #5#1_1 & \cdots & \cdots & #5^{#3}#1_{#3}\cr #2_0 & #5#2_1 & \cdots & \cdots &\cdots &\cdots &\cdots & #5^{#4}#2_{#4} & 0 & \cdots & 0\cr 0 & #2_0 & #5#2_1 & \cdots & \cdots & \cdots &\cdots &\cdots & #5^{#4}#2_{#4} & 0 &\cdots\cr \cdots &\cdots &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr 0 & \cdots & 0 & #2_0 & #5#2_1 & \cdots & \cdots&\cdots &\cdots &\cdots & #5^{#4}#2_{#4}\cr }} \def\resultanthtt#1#2#3#4#5{% \dmatrice{ #5#1_0 & #5^{2}#1_1 & \cdots & \cdots & #5^{#3+1}#1_{#3} & 0 &\cdots &\cdots &\cdots &\cdots & 0\cr 0 & #5^{2}#1_0 & #5^{3}#1_1 & \cdots & \cdots & #5^{#3+2}#1_{#3} & 0 &\cdots &\cdots &\cdots & 0\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr \cdots &\cdots &\cdots & \cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots &\cdots\cr 0 & \cdots &\cdots &\cdots &\cdots & 0 & #5^{#4}#1_0 & #5^{#4+1}#1_1 & \cdots & \cdots & #5^{#3+#4}#1_{#3}\cr #5#2_0 & #5^{2}#2_1 & \cdots & \cdots &\cdots &\cdots &\cdots & #5^{#4+1}#2_{#4} & 0 & \cdots & 0\cr 0 & #5^{2}#2_0 & #5^{3}#2_1 & \cdots & \cdots & \cdots &\cdots &\cdots & #5^{#4+2}#2_{#4} & 0 &\cdots\cr \cdots &\cdots &\cdots & \cdots &\cdots &\cdots &\cdots &\cdots & \cdots & \cdots &\cdots \cr 0 & \cdots & 0 & #5^{#3}#2_0 & #5^{#3+1}#2_1 & \cdots & \cdots&\cdots &\cdots &\cdots & #5^{#3+#4}#2_{#4}\cr }} \def\resultantb#1#2#3#4{% \dmatrice{ 1 & #1_1 & \cdots & \cdots & \cdots & #1_{#3} & 0 & \cdots &\cdots & 0\cr 0 & 1 & #1_1 & \cdots & \cdots & \cdots & #1_{#3} & 0 & \cdots&\cdots\cr \cdots &\cdots &\cdots &\cdots & \cdots & \cdots & \cdots & \cdots &\cdots &\cdots \cr 0 & \cdots & 0 & 1 & #1_1 & \cdots & \cdots&\cdots&\cdots& #1_{#3}\cr #3 & (#3-1)#2_1 & \cdots & \cdots _{#4} & 0 & \cdots & \cdots & \cdots & 0\cr 0 & #3 & (#3-1)#2_1 & \cdots &\cdots& #2_{#4} & 0 & \cdots &\cdots & \cdots & \cr \cdots &\cdots &\cdots &\cdots & \cdots &\cdots & \cdots & \cdots & \cdots &\cdots \cr 0 & \cdots & \cdots & 0 & #3 & (#3-1)#2_1 & \cdots & \cdots & \cdots& #2_{#4}\cr }} % \def\vandermonde#1#2{% \dmatrice{1& 1&\ldots& 1\cr #1_1& #1_2&\ldots& #1_{#2}\cr #1_{1}^{2}& #1_{2}^{2}&\ldots& #1_{#2}^{2}\cr \vdots&\vdots&\ddots&\vdots\cr #1_{1}^{#2 - 1}& #1_{2}^{#2 - 1}&\ldots& #1_{#2}^{#2 - 1}\cr}} % % \def\svandermonde#1#2{% \dmatrice{1& 1&\ldots& 1\cr 0& #1_2 - #1_1&\ldots& #1_{#2} - #1_1\cr 0& #1_{2}(#1_{2} - #1_1)&\ldots& #1_{#2}(#1_{#2} - #1_1)\cr \vdots&\vdots&\ddots&\vdots\cr 0& #1_{2}^{#2 - 2}(#1_{2} - #1_1)&\ldots& #1_{#2}^{#2 - 2}(#1_{#2} - #1_1)\cr}} % \def\sindeta#1#2{% \dmatrice{ \sin #2_1 & \sin #2_2 & \ldots & \sin #2_{#1}\cr -\sin^3 #2_1 & -\sin^3 #2_2 & \ldots & -\sin^3 #2_{#1}\cr \vdots&\vdots&\ddots&\vdots\cr (-1)^{#1 -1}\sin^{2#1-1} #2_1 & (-1)^{#1 -1}\sin^{2#1-1} #2_2 & \ldots & (-1)^{#1 -1}\sin^{2#1-1} #2_{#1}\cr }} \def\sindetb#1#2{% \dmatrice{ \sin #2_1 & \sin #2_2 & \ldots & \sin #2_{#1}\cr \sin 2#2_1 & \sin 2#2_2 & \ldots & \sin 2#2_{#1}\cr \vdots&\vdots&\ddots&\vdots\cr \sin #1#2_1 & \sin #1#2_2 & \ldots & \sin #1#2_{#1}\cr }} \def\sindetc#1#2{% \dmatrice{ #2_1 & #2_2 & \ldots & #2_{#1}\cr - #2_1^3 & -#2_2^3 & \ldots & -#2_{#1}^3\cr \vdots&\vdots&\ddots&\vdots\cr (-1)^{#1-1} #2_1^{2#1-1} & (-1)^{#1-1} #2_2^{2#1-1} & \ldots & (-1)^{#1-1} #2_{#1} ^{2#1-1}\cr }} % \def\castel#1#2{% \dmatrice{ 1& \sigma_1(#1_{1},\ldots,#1_{#2})&\ldots& \sigma_{#2}(#1_{1},\ldots,#1_{#2}) \cr 1& \sigma_1(#1_{2},\ldots,#1_{#2+1})&\ldots& \sigma_{#2}(#1_{2},\ldots,#1_{#2+1}) \cr 1& \sigma_1(#1_{3},\ldots,#1_{#2+2})&\ldots& \sigma_{#2}(#1_{3},\ldots,#1_{#2+2}) \cr \vdots&\vdots&\ddots&\vdots\cr 1& \sigma_1(#1_{#2+1},\ldots,#1_{2#2})&\ldots& \sigma_{#2}(#1_{#2+1},\ldots,#1_{2#2}) \cr}} % % %\def\natnums{{\bf N}} \def\natnums{{\mathbb N}} %\def\integs{{\bf Z}} \def\integs{{\mathbb Z}} %\def\rats{{\bf Q}} \def\rats{{\mathbb Q}} %\def\reals{{\bf R}} \def\reals{{\mathbb R}} %\def\complex{{\bf C}} \def\complex{{\mathbb C}} \def\Ker{{\rm Ker}\,} \def\coker{{\rm Coker}\,} \def\Kerof#1{\Ker(#1)} \let\Immag=\Im \def\Im{{\rm Im}\,} \def\Imof#1{\Im(#1)} \def\dual#1{#1^{*}} \def\dualh#1{{#1}'} \def\bdual#1{#1^{**}} \def\id{{\rm id}} \def\dimm{{\rm dim}} \def\codim{{\rm codim}} \def\rg{{\rm rk}} \def\card{{\rm card}} \def\deg#1{{\rm deg}#1} \def\mdeg{m} \def\ndeg{n} \def\pdeg{p} \def\qdeg{q} \def\ddeg{d} \def\Ndeg{N} \def\Jet{{\rm Jet}} \def\coJet{\hbox{co-Jet}} \def\gr{{\rm gr}} \def\ratio{{\rm ratio}} \def\ideal#1{\mfrac{#1}} \def\hatplus{\>\widehat{+}\>} \def\hatminus{\>\widehat{-}\>} \def\pcompl#1{\widetilde{#1}} \def\ptinf#1{{#1}_{\infty}} %\def\ptinf#1{\vector{#1}_{\infty}} \def\homog#1{#1_{*}} \def\tensalg{{\rm T}} \def\salg{{\rm S}} \def\symalg{{\rm Sym}} \def\extalg{\bigwedge} \def\domm{{\rm dom}} \def\Bezierbc#1#2#3#4#5{{\cal B}\Big[#1_{#2},\ldots,#1_{#3};\, [#4,\,#5]\Big]} \def\Bezierc#1#2#3{{\cal B}\Big[#1;\, [#2,\,#3]\Big]} %\def\prospac#1#2{{\bf P}^{#1}_{#2}} \def\prospac#1#2{{\mathbb{P}}^{#1}_{#2}} %\def\rprospac#1{{\bf RP}^{#1}} \def\rprospac#1{{\mathbb{RP}}^{#1}} %\def\cprospac#1{{\bf CP}^{#1}} \def\cprospac#1{{\mathbb{CP}}^{#1}} \def\projs#1{{\bf P}(#1)} \def\Proj{\mathrm{Proj}} \def\bprojs#1{{\bf P}\bigl(#1\bigr)} %\def\projr#1{{\bf P}^{#1}} \def\projr#1{{\mathbb{P}}^{#1}} \def\projv{\mathrm{Proj}} \def\affs{\s{E}} \def\christofa#1#2#3{[#1\, #2;\, #3]} \def\christofb#1#2#3{\Gamma_{#1\, #2}^{#3}} %\def\pairtb#1#2{\left< #1,\; #2\right>} \def\pairtb#1#2{\left< #1, #2\right>} \def\mbold#1{{\bf #1}} %\def\mfrac#1{{\EuFrak #1}} \def\mfrac#1{{\mathfrak{#1}}} \def\voldeta#1#2{% \dmatrice{ \matel{#1}{0}{1}&\matel{#1}{0}{2}&\ldots & \matel{#1}{0}{#2}& 1\cr \matel{#1}{1}{1}&\matel{#1}{1}{2}&\ldots & \matel{#1}{1}{#2}& 1\cr \vdots&\vdots&\ddots&\vdots&\vdots\cr \matel{#1}{#2}{1}&\matel{#1}{#2}{2}&\ldots& \matel{#1}{#2}{#2}& 1\cr}} % \def\voldetb#1#2{% \dmatrice{ \matel{#1}{1}{1} - \matel{#1}{0}{1}&\matel{#1}{1}{2} - \matel{#1}{0}{2}& \ldots & \matel{#1}{1}{#2} - \matel{#1}{0}{#2}\cr \matel{#1}{2}{1} - \matel{#1}{0}{1}&\matel{#1}{2}{2} - \matel{#1}{0}{2}& \ldots & \matel{#1}{2}{#2} - \matel{#1}{0}{#2}\cr \vdots&\vdots&\ddots&\vdots\cr \matel{#1}{#2}{1} - \matel{#1}{0}{1}&\matel{#1}{#2}{2} - \matel{#1}{0}{2}& \ldots&\matel{#1}{#2}{#2} - \matel{#1}{0}{#2}\cr}} % \def\gramdet#1#2{% \dmatrice{ \norme{#1_1}^2 &\pairt{#1_1}{#1_2}&\ldots & \pairt{#1_1}{#1_{#2}}\cr \pairt{#1_2}{#1_1} &\norme{#1_2}^2&\ldots & \pairt{#1_2}{#1_{#2}}\cr \vdots&\vdots&\ddots&\vdots\cr \pairt{#1_{#2}}{#1_1} &\pairt{#1_{#2}}{#1_2}&\ldots & \norme{#1_{#2}}^2\cr }} % \def\chull#1{\s{C}(#1)}