How to define a multigraded ring

[QJ L]

I want to define a ring K[u,v,w].Here u has grade (1,0,0), v has grade (0,1,0), w has grade (0,0,1). How to realized it? 

[Dima Pasechnik]

On February 26, 2026 7:49:46 PM CST, QJ L <s3456...@gmail.com> wrote:
>I want to define a ring K[u,v,w].Here u has grade (1,0,0), v has grade
>(0,1,0), w has grade (0,0,1). How to realized it?
>

It's not clear to me which monomial ordering you have in mind.
Can you consult <https://www.singular.uni-kl.de/Manual/4-4/sing_28.htm#SEC39>
(Sage more or less follows Singular in the part)
and be more precise about the grading you want?

[QJ L]

I want to define a coordinate ring of P1*P1*P3, i.e. K[u1,u2,v1,v2,w1,w2,w3,w4] .Here u1,u2 has grade (1,0,0). v1,v2 has grade (0,1,0). w1,w2,w3,w4 has grade (0,0,1). As for the ordering, I have no specific choice, maybe just the naive choice u1<u2, v1<v2,w1<w2<w3<w4,(1,0,0)<(0,1,0)<(0,0,1).