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Sage 10.3.beta8:
```sage
sage: R. = GF(11)[]
sage: I = [x^2+1, x+y]
sage: J = [x]
sage: 1 in R.ideal(I + J)
True
sage: 1 in R.quotient(I).ideal(J)
False
```
See also #33217.
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This is from SO:
https://stackoverflow.com/questions/72620435/complete-solutions-of-system-of-equations-with-python/72635743?noredirect=1#comment128326052_72635743
The following system of polynomi…
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Trying to compute a Gröbner fan over a ring in which one variable name contains the other as a prefix gives the following error:
```
sage: P = PolynomialRing(QQ,3*5,"x"); x = P.gens(); M = Matrix(3…
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Let A = (a_{ij}) be an m x n (m
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The dimension of an ideal, that is the Krull dimension of the quotient `R/I`, does not depend on the monomial order. But
```
sage: P. = PolynomialRing(QQ,order='neglex')
sage: P.ideal(x).dimension…
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Right now, a user who never used libsing before is completely left alone. The test cases are their only way to find out anything about how to use libsing, and they are not very good for that -- no exp…
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We discovered in #33134 that `hilbert_numerator()` can return different results with `algorithm="singular"` and the default `algorithm="sage"`:
```
sage: n=4; m=11; P = PolynomialRing(QQ,n*m,"x")
s…
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It seems that `neglex` used there isn't the right order, for it's not even a Groebner basis order. See
https://ask.sagemath.org/question/51851/toric-ideal-of-point-configuration-yielding-whole-ring…
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Since #3013 was merged several pieces of code I use have all ground to a halt. Previously, a change I introduced in #2766 caused `submatrix` to call `reduce`, which slowed down submatrix when the targ…