How can I convert a univariate polynomial into a multivariate one? The only idea I have is introducing a dummy (silent) variable, but is there a more straightforward way?
PS. The reason for using such a conversion is that the multivariate polynomial ring has some methods (such as .reduce()) that are unavailable for univariate polynomials.
Sage has both
Hopefully multivariate poylnomial rings in a single variable behave as you want.
Example.
Say we have defined
sage: x = polygen(QQ)
sage: f = x^2 + x + 1
sage: R = f.parent()
or
sage: R.<x> = QQ[]
sage: f = x^2 + x + 1
There are various ways to define the multivariate counterpart to f.
Some ways straight from f:
sage: g = PolynomialRing(f.base_ring(), 1, str(f.variables()[0]))(f)
sage: g = PolynomialRing(f.base_ring(), 1, str(f.parent().gen()))(f)
sage: g = PolynomialRing(f.base_ring(), 1, str(parent(f).gen()))(f)
sage: g = PolynomialRing(f.base_ring(), 1, f.variable_name())(f)
sage: g
x^2 + x + 1
sage: parent(g)
Multivariate Polynomial Ring in x over Rational Field
Some ways using R:
sage: M = PolynomialRing(R.base_ring(), 1, str(R.gen()))
sage: M = PolynomialRing(R.base_ring(), 1, R.variable_name())
sage: g = M(f)
Helper function.
As you suggest, similar to g.univariate_polynomial(),
one could expect f.multivariate_polynomial().
It might be worth opening a ticket to add such a method.
In the meantime, one can use a helper function:
def multivariate(f):
r"""
Return this polynomial in one variable as an element
in a multivariate polynomial ring with one variable.
EXAMPLES::
sage: x = polygen(QQ)
sage: f = x^2 + x + 1
sage: parent(f)
Univariate Polynomial Ring in x over Rational Field
sage: g = multivariate(f)
sage: parent(g)
Multivariate Polynomial Ring in x over Rational Field
sage: g
x^2 + x + 1
sage: f == g and g == f and g.univariate_polynomial() == f
True
"""
R = f.parent()
if str(R).startswith('Univariate'):
R = PolynomialRing(R.base_ring(), 1, R.variable_name())
f = R(f)
return f
Yes, I have found it. The conversion is still looks ugly though. I was hoping for a method similar to .univariate_polynomial() doing the opposite conversion.
I've requested the .multivariate_polynomial()method in https://github.com/sagemath/sage/issu...
Let say we have
R.<x> = QQ[]
f = x^2 + x + 1
My ugly solution is a conversion:
g = PolynomialRing( f.base_ring(), 2, str(f.variables()[0]) + ',dummy' )(f)
for which we have:
print(g.parent())
Multivariate Polynomial Ring in x, dummy over Rational Field
Why not just start with
R.<x,dummy> = QQ[]
and then any polynomials you later construct in x are automatically multivariate? I must be missing something about the workflow here.
It was just an example. In the actual application the polynomial comes from elsewhere and I need to convert it.
Hence the very first question I asked: "How are you defining your polynomials in the first place?"
Asked: 2023-08-17 19:20:47 +0100
Seen: 216 times
Last updated: Aug 18 '23
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How are you defining your polynomials in the first place? The dummy variable seems pretty straightforward to me: just set
R.<x,y> = ZZ[]and do all of your computations withx. Alternatively find better alternatives (likequo_reminstead ofreduce?) in the single variable case.Let's assume that polynomial comes from some function and (for simplicity) it's defined as an element of
QQ['x']. The functionquo_remtakes a single argument whilereducecan do the job over multiple given polynomials. But most importantly, I don't want to make distinction between univariate and multivariate polynomials but rather treat the former as latter.To make sure that a polynomial
fhas the correct parent and therefore the desired methods, I don't see a more straightforward way than defining a multivariate polynomial ring and definingxto be one of the generators: after one or two lines of setup at the beginning, you are working the context you want.