The leading coefficient of a resultant

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Let $k$ be a field and $f,gin k[x,y]$. Let $R(y)=Res_x(f,g)$ be the resultant of $f$ and $g$ considered as polynomials in x over the field $k(y)$. Can you give a nice formula for the leading coefficient of $R(y)$?



Since I am interested only in the leading coefficient of $R(y)$ I was wondering if one could replace $(f,g)$ with $(F,G)$, where F and G are actually "simpler" polynomials in such a way that $Res_x(f,g)=Res_x(F,G)$ (of course, a more explicit formula would be even better).










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    Let $k$ be a field and $f,gin k[x,y]$. Let $R(y)=Res_x(f,g)$ be the resultant of $f$ and $g$ considered as polynomials in x over the field $k(y)$. Can you give a nice formula for the leading coefficient of $R(y)$?



    Since I am interested only in the leading coefficient of $R(y)$ I was wondering if one could replace $(f,g)$ with $(F,G)$, where F and G are actually "simpler" polynomials in such a way that $Res_x(f,g)=Res_x(F,G)$ (of course, a more explicit formula would be even better).










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      up vote
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      down vote

      favorite











      Let $k$ be a field and $f,gin k[x,y]$. Let $R(y)=Res_x(f,g)$ be the resultant of $f$ and $g$ considered as polynomials in x over the field $k(y)$. Can you give a nice formula for the leading coefficient of $R(y)$?



      Since I am interested only in the leading coefficient of $R(y)$ I was wondering if one could replace $(f,g)$ with $(F,G)$, where F and G are actually "simpler" polynomials in such a way that $Res_x(f,g)=Res_x(F,G)$ (of course, a more explicit formula would be even better).










      share|cite|improve this question













      Let $k$ be a field and $f,gin k[x,y]$. Let $R(y)=Res_x(f,g)$ be the resultant of $f$ and $g$ considered as polynomials in x over the field $k(y)$. Can you give a nice formula for the leading coefficient of $R(y)$?



      Since I am interested only in the leading coefficient of $R(y)$ I was wondering if one could replace $(f,g)$ with $(F,G)$, where F and G are actually "simpler" polynomials in such a way that $Res_x(f,g)=Res_x(F,G)$ (of course, a more explicit formula would be even better).







      polynomials resultant






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      asked Sep 1 at 9:52









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