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Fix loss of precision from singular in multivariable polynomial rings #39651

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Fixes #39640. As pointed out in #39640, there can be a drastic loss of precision when a polynomial over an inexact real field is processed by singular, and the cause seems to be an insufficient number of "stability bits" in the singular base field. So this PR changes the number of stability bits from 0 to 10 in the singular interface.

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nbruin commented Mar 8, 2025

Good find! It looks like singular basically uses a form of ball or interval arithmetic. So if we're coming from a real field in sage that doesn't have that info, we'd need to choose how to set the error. I think the only choice you have is to assume that pretty much all digits are significant. Surely numbers from RealField(200) should carry about twice as many significant digits as RealField(100).

We have RealBallField and the interval fields. From those, we could actually take the right cues to get an error bound.

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loss of digits in multivariate polynomial ring quotients with base ring RR
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