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.
On Wednesday, 5 March 2025 at 10:51:29 UTC-8 Seth Chaiken 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.
>
> 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.
>
>
>
>
>
>
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