Resultant of two plane forms, viewed in one variable

Let $k$ be a field and let $F,G\in k[x_0,x_1,x_2]$ be nonzero homogeneous forms of positive total degrees $d,e$ homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$. Write $F={\sum }_{i=0}^d{a}_i(x_0,x_1)x_2^i$, $G={\sum }_{i=0}^e{b}_i(x_0,x_1)x_2^i$. Over $R=k[x_0,x_1]$ introduce auxiliary variables $X,Y$ and homogenise with the nominated degrees:

$H_F(X,Y)={\sum }_{i=0}^d{a}_i{X}^i{Y}^{d-i},H_G(X,Y)={\sum }_{i=0}^e{b}_i{X}^i{Y}^{e-i}\mathrm{.}$

Define the resultant eliminating $x_2$ by

${\mathrm{Res}}_{x_2}(F,G):={\mathrm{Res}}_{d,e}(H_F,H_G)\in k[x_0,x_1],$

using the Sylvester determinant and ordered bases of Sylvester resultant of two positive-degree binary forms. Thus $H_F(X,1)=F(x_0,x_1,X)$, but the nominations remain $d,e$ even if the actual degrees in $X$ drop.

The resultant is zero or homogeneous of total degree $de$ in $x_0,x_1$. To see this, index matrix rows by the exponent $r=0,\dots ,d+e-1$ of $X$ in the target monomial, and index the $F$-columns by multiplier exponents $j=0,\dots ,e-1$ and the $G$-columns by $j=0,\dots ,d-1$. An $F$-entry is $a_{r-j}$ of degree $d-r+j$ and a $G$-entry is $b_{r-j}$ of degree $e-r+j$. Every nonzero determinant term therefore has total degree

$ed+de+\frac{e(e-1)}{2}+\frac{d(d-1)}{2}-\frac{(d+e-1)(d+e)}{2}=de\mathrm{.}$

Specialisation. Coefficient specialisation sends the determinant to the determinant of $H_F(a,b;X,Y)$ and $H_G(a,b;X,Y)$, for any $(a,b)\mathrm{\ne }(0,0)$ Scaling, specialization, and the affine and infinite charts of a binary resultant. Over an algebraically closed extension, its vanishing detects a common projective root of these binary forms The binary Sylvester resultant detects a common geometric projective root. Such a root is either $[c:1]$, with $F(a,b,c)=G(a,b,c)=0$, or $[1:0]$, when both coefficients $a_d,b_e$ vanish. Consequently the value need not detect a finite root when both leading coefficients vanish. The later detection lemma excludes that case by assuming $[0:0:1]$ lies on neither curve.

• The elimination coordinate is fixed; the resultant polynomial depends on it. The underlying projective curves do not depend on a choice of coordinates.
• The scaling rule gives ${\mathrm{Res}}_{x_2}(uF,vG)=u^e{v}^d{\mathrm{Res}}_{x_2}(F,G)$ for $u,v\in k$ Scaling, specialization, and the affine and infinite charts of a binary resultant. In particular it is not a scalar-independent function of the curves.