Multivariate polynomial ring with total degree no larger than n

timaeus
15 ●1 ●5

slelievre
17789 ●22 ●163 ●352 http://carva.org/samue...

Hello everyone,

I need to compute the product: $f(x_1) f(x_2) ... f(x_n)$ , where $f$ is a polynomial of degree $n$ , and I do not need the part with total degree larger than $n$ . To reduce the computation complexity, I think it would be helpful to construct an $n$ -variable multivariate polynomial ring, with terms total degree no larger than $n$ . I found the following two possible ways to do this, however, I could not make either of them work for my settings.

Create a multivariate polynomial ring then quotient out every monomial with total degree larger than $n$ . However I do not know how to express this ideal.
```
Q = PolynomialRing(QQ, n, 'x') 
x = Q.gens()
```
I found a suggestion to use a function max_degree for polynomial rings at
- Ask Sage question 55842: Set of polynomials under a certain degree
However, it seems there is no max_total_degree function for the multivariate case.

I am new to Python, Sage, and this community, so thank you in advance for your helpful suggestions and comments!

Comments

Welcome to Ask Sage! Thank you for your question.

slelievre ( 3 years ago )

To post links as a new user, insert spaces in them. Someone can then fix them.

Example: https ://doc .sagemath .org

slelievre ( 3 years ago )

Thank you very much for fixing my link and your comment!

timaeus ( 3 years ago )

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answered 3 years ago

Max Alekseyev

6932 ●9 ●52 ●162

updated 3 years ago

Ignoring parts with total degree $\geq d$ suggests that power series with precision $d$ are more suitable in this setting:

d = 3
Q = PowerSeriesRing(QQ, 3, 'x', default_prec=d) 
x = [g.add_bigoh(d) for g in Q.gens()]
f = x[0]^4
print( f.add_bigoh(d).polynomial() )

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Comments

Thank you for your answer! I tried the following code: Q = PowerSeriesRing(QQ, 3, 'x', default_prec = 3); x = Q.gens(); f = x[0]^2; print(f^3) However I still get x0^6 here instead of an expected 0.

timaeus ( 3 years ago )

It turns out generators do not inherit default precision, so it needs to be added explicitly. In your example, it would be x = [g.add_bigoh(3) for g in Q.gens()].

Max Alekseyev ( 3 years ago )

P = PowerSeriesRing(QQ, 3, 'x', default_prec = 3); x = [g.add_bigoh(3) for g in P.gens()]; f=P(x[0]^4); f

The above code still yields x0^4 + O(x0, x1, x2)^6 instead of 0. Also, since I need to convert the product into symmetric polynomials afterwards, the introducing of the big O seems cause more trouble...

timaeus ( 3 years ago )

The reason is that Sage tries to keep precision as large as possible, and for the 4th power the precision becomes 6. It can be always decreased by .add_bigoh() and the resulting series can be converted into a polynomial via .polynomial() method. I've updated an example in my answer accordingly.

Max Alekseyev ( 3 years ago )

Thank you very much for your answer! It works fine now. By the way, do you know how to fix this problem under your settings?

g(y) = y^2; g(x[0])

This code used to work when Q is a polynomial ring, but it seems get some trouble when Q is a power series ring.

timaeus ( 3 years ago )

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