The function call IteratedResultantDim0(f, rc, R) returns the iterated resultant of f w.r.t. rc. See the command IteratedResultant for a definition of the notion of an iterated resultant.

rc is assumed to be a zero-dimensional normalized 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\;$$\mathrm{vars}≔\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\:$$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\,p\right)\:$

$p≔962592769$

$N≔\mathrm{nops}\left(\mathrm{vars}\right)\:$$\mathrm{dg}≔3\:$$\mathrm{degs}≔\left[\mathrm{seq}\left(4\,i=1..N\right)\right]\:$$\mathrm{pol}≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\;$$\mathrm{tc}≔\mathrm{RandomRegularChainDim0}\left(\mathrm{vars}\,\mathrm{degs}\,p\right)\;$$\mathrm{Equations}\left(\mathrm{tc}\,R\right)$

$\mathrm{pol}≔962592762{\mathrm{x1}}^3+22{\mathrm{x1}}^2\mathrm{x2}+962592714{\mathrm{x1}}^2\mathrm{x3}+962592675{\mathrm{x1}}^2\mathrm{x4}+962592713\mathrm{x1}{\mathrm{x2}}^2+962592707\mathrm{x1}\mathrm{x2}\mathrm{x4}+962592696\mathrm{x1}{\mathrm{x3}}^2+962592765\mathrm{x1}\mathrm{x3}\mathrm{x4}+962592759\mathrm{x1}{\mathrm{x4}}^2+80{\mathrm{x2}}^3+962592725{\mathrm{x2}}^2\mathrm{x3}+71{\mathrm{x2}}^2\mathrm{x4}+962592694\mathrm{x2}{\mathrm{x3}}^2+962592759\mathrm{x2}\mathrm{x3}\mathrm{x4}+962592729\mathrm{x2}{\mathrm{x4}}^2+23{\mathrm{x3}}^3+75{\mathrm{x3}}^2\mathrm{x4}+6\mathrm{x3}{\mathrm{x4}}^2+37{\mathrm{x4}}^3+87{\mathrm{x1}}^2+97\mathrm{x1}\mathrm{x2}+962592686\mathrm{x1}\mathrm{x3}+62\mathrm{x1}\mathrm{x4}+962592752{\mathrm{x2}}^2+962592762\mathrm{x2}\mathrm{x3}+42\mathrm{x2}\mathrm{x4}+962592677{\mathrm{x3}}^2+74\mathrm{x3}\mathrm{x4}+962592746{\mathrm{x4}}^2+962592687\mathrm{x1}+962592719\mathrm{x2}+72\mathrm{x3}+87\mathrm{x4}+874547123$

$\mathrm{tc}≔\mathrm{regular\_chain}$

$\left[{\mathrm{x1}}^4+962592759{\mathrm{x1}}^3+\left(962592687\mathrm{x2}+71\mathrm{x3}+16\mathrm{x4}+83\right){\mathrm{x1}}^2+\left(9{\mathrm{x2}}^2+\left(962592709\mathrm{x3}+962592686\mathrm{x4}+98\right)\mathrm{x2}+962592721{\mathrm{x3}}^2+\left(962592750\mathrm{x4}+62\right)\mathrm{x3}+37{\mathrm{x4}}^2+5\mathrm{x4}+96\right)\mathrm{x1}+962592752{\mathrm{x2}}^3+\left(25\mathrm{x3}+91\mathrm{x4}\right){\mathrm{x2}}^2+\left(98{\mathrm{x3}}^2+\left(962592705\mathrm{x4}+64\right)\mathrm{x3}+962592679{\mathrm{x4}}^2+962592709\mathrm{x4}+962592735\right)\mathrm{x2}+962592756{\mathrm{x3}}^3+\left(44\mathrm{x4}+962592767\right){\mathrm{x3}}^2+\left(71{\mathrm{x4}}^2+962592722\mathrm{x4}+962592730\right)\mathrm{x3}+962592716{\mathrm{x4}}^3+962592697{\mathrm{x4}}^2+962592672\mathrm{x4}+91831581\,{\mathrm{x2}}^4+{\mathrm{x2}}^3+\left(\mathrm{x3}+55\mathrm{x4}+962592741\right){\mathrm{x2}}^2+\left(16{\mathrm{x3}}^2+\left(30\mathrm{x4}+962592742\right)\mathrm{x3}+962592754{\mathrm{x4}}^2+962592710\mathrm{x4}+962592673\right)\mathrm{x2}+72{\mathrm{x3}}^3+\left(962592682\mathrm{x4}+47\right){\mathrm{x3}}^2+\left(962592679{\mathrm{x4}}^2+43\mathrm{x4}+92\right)\mathrm{x3}+962592678{\mathrm{x4}}^3+962592681{\mathrm{x4}}^2+962592721\mathrm{x4}+614095058\,{\mathrm{x3}}^4+11{\mathrm{x3}}^3+\left(962592720\mathrm{x4}+962592722\right){\mathrm{x3}}^2+\left(40{\mathrm{x4}}^2+962592688\mathrm{x4}+91\right)\mathrm{x3}+68{\mathrm{x4}}^3+962592759{\mathrm{x4}}^2+31\mathrm{x4}+175602554\,{\mathrm{x4}}^4+962592746{\mathrm{x4}}^3+10{\mathrm{x4}}^2+962592708\mathrm{x4}+685457535\right]$

$\mathrm{r1}≔\mathrm{IteratedResultantDim0}\left(\mathrm{pol}\,\mathrm{tc}\,R\right)$

$\mathrm{r1}≔446889812$

$\mathrm{r2}≔\mathrm{IteratedResultant}\left(\mathrm{pol}\,\mathrm{tc}\,R\right)$

$\mathrm{r2}≔446889812$

$\mathrm{Expand}\left(\mathrm{r1}-\mathrm{r2}\right)\mathbf{mod}p$

$0$