The command NormalizePolynomialDim0 returns a normalized form of f w.r.t. rc, that is, a polynomial $q$ which is associated to f modulo rc, such that $q$ is normalized w.r.t. rc.

rc is zero-dimensional regular chain, and f together with rc forms a zero-dimensional regular chain.

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

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

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

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

$p≔962592769\:$

$\mathrm{vars}≔\left[y\,x\right]\:$

$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\,p\right)\:$

$\mathrm{f1}≔x\left(y^2+y+1\right)+2\:$

$\mathrm{f2}≔\left(x+1\right)\left(y^2+y+1\right)+x^3+x+1\:$

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

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

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

$\mathrm{r2}≔x^8+2x^6+962592767x^5+962592766x^4+962592767x^3+962592766x^2+4x+4$

$\mathrm{Gcd}\left(\mathrm{r2}\,x\left(x+1\right)\right)\mathbf{mod}p$

$1$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\mathrm{g2}≔\left[\left[x^3+x{y}^2+xy+y^2+2x+y+2\,\mathrm{regular\_chain}\right]\,\left[x^3+x{y}^2+xy+y^2+2x+y+2\,\mathrm{regular\_chain}\right]\right]$

$\mathrm{NormalizePolynomialDim0}\left(\mathrm{g2}\left[1\right]\left[1\right]\,\mathrm{g2}\left[1\right]\left[2\right]\,R\right)$

$x^3+y^2+x+y$