About Grobner basis
Suppose we have a liste of polynomials L in Z[x,y,z,w] ; I need to transform the following code in Maple to Sage
g1:=Basis(L,plex(x,y,z,w)):
isolve(g1[1]);
g2:=Basis(L,plex(z,w,x,y)):
isolve(g2[1]);
g3:=Basis(L,plex(w,y,z,x)):
isolve(g3[1]);
i.e. How to compute the grobner basis for the ideal generated by the list of polynomials L with respect to the order x >y > z > w and then with respect to the order z > w > x > y and finally with respect to the order w > y > z > x .
For example suppose,
L=[1406420415531174980327303290571117064276422184073371260430004727814077316258636748578133791670129*x^2 - 585580311734997207006635656684084836378829142925094427420106*x*w + 60953378112431066706921*w^2,
-1688076693759028688508053956627434893445056753313490532106912447426325421979583194817700061681593440864404472*x^2 + 242035925986271489335186642003579544694017559808139175285991673034916861*x*w + 22772877973564833308423102217214551*w^2,
1406420415531174980327303290571117064276422184073371260430004727814077316258636748578133791670129*x^2*y - 585580311734997207006635656684084836378829142925094427420106*x*y*w + 60953378112431066706921*y*w^2,
-1688076693759028688508053956627434893445056753313490532106912447426325421979583194817700061681593440864404472*x^2*y + 242035925986271489335186642003579544694017559808139175285991673034916861*x*y*w + 22772877973564833308423102217214551*y*w^2,
546492971093875818200311226757760151716020675484639718374766505281329940973795540404028891234781638071786218379709560581*x*y*z - 113769510467918507941520256906804857756761200776076799074471649913620059848042913417*y*z*w]
What are g1[1], g2[1] and g3[1] in your example?
When the number of solutions is finite, you can get them by calling
.variety()method of the ideal. Unfortunately, this method is defined only over the fields, but not just rings. You can try to get solutions overQQand then extract those that are inZZ.