Bug Polynomial Quotient with 2 ideal generator rings over RR lose precision of base ring.

[Seth Chaiken]

In Sage 10.4, we tried to inject a RR number into a RR polynomial ring quotient.  It

works fine when the ideal had one (monomial) generator but precision is lost when

it had two generators. 

One generator example (works):

eps = var("eps")
edBaseRing=PolynomialRing(RR,[eps])
edIdeal=ideal(edBaseRing, [eps])
edRing=edBaseRing.quotient_ring(edIdeal)

print(RR(pi))
print(edBaseRing(pi))
print(edRing(pi))

yields:

3.14159265358979 

3.14159265358979 

3.14159265358979

But with 2 generators, injecting RR(pi) into the quotient loses precision.

The analogous output is:

3.14159265358979 

3.14159265358979 

3.14200000000000

Here's the full code:

eps, dlt = var("eps,dlt")
edBaseRing=PolynomialRing(RR,[eps, dlt])
edIdeal=ideal(edBaseRing, [eps, dlt])
edRing=edBaseRing.quotient_ring(edIdeal)

print(RR(pi))
print(edBaseRing(pi))
print(edRing(pi))

Looks like a bug! I isolated it from more complex code where the ideal was simply all multiples of the two variables, and a RealField with non-default bit precision was used.

[Dima Pasechnik]

On Wed, Mar 5, 2025 at 12:51 PM Seth Chaiken <scha...@albany.edu> wrote:
>
> In Sage 10.4, we tried to inject a RR number into a RR polynomial ring quotient. It
> works fine when the ideal had one (monomial) generator but precision is lost when
> it had two generators.

It's not a bug - it's half-expected that any nontrivial algebraic
computations in PolynomialRing(RR)
will fail due to rounding errors/loss of precision.
You can increase the precision of the ring, and try again, but I'd
recommend avoiding this if at all possible.
Use exact rings instead. E.g. approximate your data by rationals and
use PolynomialRing(QQ).

HTH
Dima

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[Chaiken, Seth D]

Thanks for your reply! I had done just what you recommended.
I then coded a loop to successively take off leading polynomial terms and,
for each one, print the floating point approximation of the rational polynomial.
Seth

________________________________________
From: sage-s...@googlegroups.com <sage-s...@googlegroups.com> on behalf of Dima Pasechnik <dim...@gmail.com>
Sent: Wednesday, March 5, 2025 3:38 PM
To: sage-s...@googlegroups.com
Subject: Re: [sage-support] Bug Polynomial Quotient with 2 ideal generator rings over RR lose precision of base ring.

HTH
Dima

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[Nils Bruin]

I definitely wouldn't expect nontrivial polynomial arithmetic to work reliably over RR without special measures. There are definitely methods for algebraic geometry with floats, but it needs very special attention to numerical stability. That is already true for linear algebra over RR and for polynomial algebra it is even worse.

I am pretty sure sage does NOT have facilities for this at the moment. Even for linear algebra, I wouldn't trust that sage would select a numerically stable approach over RR. There is a good chance it will hit just generic implementations that are written for exact linear algebra.

However, the computations you are doing would trigger no polynomial arithmetic at all, so the "loss of precision" is  very suspect. It's also very suspect that it ends up with a number that has a very precise number of decimal digits. Indeed, replacing "pi" with something else, such as RR(1/3) or sqrt(2) gives the exact same rounding.

I suspect you're hitting code that somehow tries to call the Singular library and, since that doesn't support floats, it looks like it takes a very naive approximation

Integers()(a*1000)/1000. That DOES look like a bug. An error would be more informative here.

It makes sense this would only happen in a multivariate polynomial ring (regardless of what ideal you're actually using): univariate polynomial rings would just use euclidean division and not rely on the singular library. That would still be numerically unstable in many cases, but in a trivial case such as this one, that would not be an issue.

[Chaiken, Seth D]

Revised tests are consistent with Nils' reply. I replaced pi by 10/3. I compared the quotient by one variable of the
two variable ring with a quotient by the one variable in the one variable ring and got the results below, the first
is consistent with the naive approx. Integers()(a*1000)/1000:

3.33300000000000
versus
3.33333333333333

eps = var("eps")
edBaseRing=PolynomialRing(RR,[eps])
edIdeal=ideal(edBaseRing, [eps])
edRing=edBaseRing.quotient_ring(edIdeal)

rat = 10/3
print(RR(rat))
print(edBaseRing(rat))
print(edRing(rat))

3.33333333333333
3.33333333333333
3.33333333333333

eps, dlt = var("eps,dlt")
edBaseRing=PolynomialRing(RR,[eps, dlt])

#edIdeal=ideal(edBaseRing, [eps, dlt])
edIdeal=ideal(edBaseRing, [eps])
edRing=edBaseRing.quotient_ring(edIdeal)

print(RR(rat))
print(edBaseRing(rat))
print(edRing(rat))

3.33333333333333
3.33333333333333
3.33300000000000

________________________________________
From: sage-s...@googlegroups.com <sage-s...@googlegroups.com> on behalf of Nils Bruin <nbr...@sfu.ca>
Sent: Wednesday, March 5, 2025 11:49 PM
To: sage-support
Subject: [sage-support] Re: Bug Polynomial Quotient with 2 ideal generator rings over RR lose precision of base ring.

One generator example (works):

yields:
3.14159265358979
3.14159265358979
3.14159265358979

print(RR(pi))
print(edBaseRing(pi))
print(edRing(pi))

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[Nils Bruin]

This is now https://github.com/sagemath/sage/issues/39640