The command BivariateModularTriangularize(F, R) returns a triangular decomposition of F in R. See the command Triangularize and the page RegularChains for the concept of a triangular decomposition.

F consists of two bivariate polynomials f1 and f2 of R. No other assumptions are required.

R must have only two variables and no parameters.

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 f1 and f2 are, the larger must be $e$ such that $2^e$ divides $p-1$.  If the degree of  f1 or f2 is too large, then an error is raised.

The algorithm is deterministic (i.e. non-probabilistic) and uses modular techniques together with asymptotically fast polynomial arithmetic.

When both Triangularize and BivariateModularTriangularize apply, the latter command is very likely to outperform the former one.

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

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

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

$p≔469762049\;$$\mathrm{vars}≔\left[x\,y\right]\;$$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\,p\right)$

$p≔469762049$

$\mathrm{vars}≔\left[x\,y\right]$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{f1}≔x^{49}+y+1$

$\mathrm{f1}≔x^{49}+y+1$

$\mathrm{f2}≔x+y^{41}+1$

$\mathrm{f2}≔y^{41}+x+1$

$\mathrm{dec}≔\mathrm{BivariateModularTriangularize}\left(\left[\mathrm{f1}\,\mathrm{f2}\right]\,R\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

$\left[\left[x+y^{41}+1\,y^{2009}+49y^{1968}+1176y^{1927}+18424y^{1886}+211876y^{1845}+1906884y^{1804}+13983816y^{1763}+85900584y^{1722}+450978066y^{1681}+175407438y^{1640}+231867703y^{1599}+10669226y^{1558}+190373232y^{1517}+469560422y^{1476}+200808123y^{1435}+468552287y^{1394}+114869768y^{1353}+112450244y^{1312}+460890461y^{1271}+133871214y^{1230}+200806821y^{1189}+8869201y^{1148}+11288074y^{1107}+319617771y^{1066}+228812073y^{1025}+228812073y^{984}+319617771y^{943}+11288074y^{902}+8869201y^{861}+200806821y^{820}+133871214y^{779}+460890461y^{738}+112450244y^{697}+114869768y^{656}+468552287y^{615}+200808123y^{574}+469560422y^{533}+190373232y^{492}+10669226y^{451}+231867703y^{410}+175407438y^{369}+450978066y^{328}+85900584y^{287}+13983816y^{246}+1906884y^{205}+211876y^{164}+18424y^{123}+1176y^{82}+49y^{41}+469762048y\right]\right]$

$\mathrm{map}\left(\mathrm{NumberOfSolutions}\,\mathrm{dec}\,R\right)$

$\left[2009\right]$