The call ResultantBySpecializationCube(f1, f2, v, SCube, R) returns the resultant of f1 and f2 w.r.t. v. It is computed by interpolating the data in SCube. See the command SubresultantChainSpecializationCube to learn how to build this data-structure.

f1 and f2 must have main variable v and $\mathrm{degree}\left(\mathrm{f2}\,v\right)\le \mathrm{degree}\left(\mathrm{f1}\,v\right)$ must hold.

R must have a prime characteristic $p$ such that FFT-based polynomial arithmetic can be used for this computation. The higher the degrees of f1 and f2 are, the larger $e$ must be such that $2^e$ divides $p-1$.  If the degree of  f1 or f2 is too large, then an error is raised.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{FastArithmeticTools}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$p≔962592769\;$$\mathrm{vars}≔\left[x\,a\,b\,c\,d\right]\;$$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\,p\right)$

$p≔962592769$

$\mathrm{vars}≔\left[x\,a\,b\,c\,d\right]$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{f1}≔\left(x^2-a\right)\left(x-b\right)\mathbf{mod}p$

$\mathrm{f1}≔\left(x^2+962592768a\right)\left(x+962592768b\right)$

$\mathrm{f2}≔\left(x^2-c\right)\left(x-d\right)\mathbf{mod}p$

$\mathrm{f2}≔\left(x^2+962592768c\right)\left(x+962592768d\right)$

$\mathrm{SCube}≔\mathrm{SubresultantChainSpecializationCube}\left(\mathrm{f1}\,\mathrm{f2}\,x\,R\,1\right)$

$\mathrm{SCube}≔\mathrm{subresultant\_chain\_specialization\_cube}$

$\mathrm{r2}≔\mathrm{ResultantBySpecializationCube}\left(\mathrm{f1}\,\mathrm{f2}\,x\,\mathrm{SCube}\,R\right)$

$\mathrm{r2}≔a^2{b}^3{d}^2+962592768a^2{b}^2{d}^3+962592767a{b}^{3}c{d}^2+2a{b}^{2}c{d}^3+d^2{c}^2{b}^3+962592768d^3{c}^2{b}^2+962592768a^3{b}^3+a^3{b}^{2}d+2a^2{b}^{3}c+962592767a^2{b}^{2}cd+962592768a^{2}bc{d}^2+a^{2}c{d}^3+962592768a{b}^3{c}^2+a{b}^2{c}^{2}d+2ab{c}^2{d}^2+962592767a{c}^2{d}^3+962592768d^2{c}^{3}b+d^3{c}^3+a^{3}bc+962592768a^{3}cd+962592767a^{2}b{c}^2+2a^2{c}^{2}d+ab{c}^3+962592768a{c}^{3}d$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[\mathrm{r2}\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{g2}≔\mathrm{RegularGcdBySpecializationCube}\left(\mathrm{f1}\,\mathrm{f2}\,\mathrm{rc}\,\mathrm{SCube}\,R\right)$

$\mathrm{g2}≔\left[\left[962592768a{b}^{2}d+a{b}^{2}x+ab{d}^2+962592768a{d}^{2}x+cd{b}^2+962592768b^{2}cx+962592768d^{2}cb+c{d}^{2}x+962592768b{a}^2+a^{2}x+abc+acd+962592767acx+962592768d{c}^2+c^{2}x\,\mathrm{regular\_chain}\right]\,\left[b{x}^2+962592768d{x}^2+962592768ab+ax+cd+962592768cx\,\mathrm{regular\_chain}\right]\,\left[b{x}^2+962592768d{x}^2+962592768ab+ax+cd+962592768cx\,\mathrm{regular\_chain}\right]\right]$