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### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex') sage: f1=x1+x2+x3+x4+x5 sage: f2=x1x2+x2x3+x3x4+x4x5+x1x5 sage: f3=x1x2x3+x2x3x4+x3x4x5+x4x5x1+x5x1x2 sage: f4=x1x2x3x4+x2x3x4x5+x3x4x5x1+x4x5x1x2+x5x1x2x3 sage: f5=x1x2x3x4x5-1 sage: I = Ideal(f1,f2,f3,f4,f5) sage: I.variety()

produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation. [{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]

which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level of the output but without success.

Thanks for your time!

### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex') sage: f1=x1+x2+x3+x4+x5 PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1x2+x2x3+x3x4+x4x5+x1x5 x5+x1*x5

sage: f3=x1x2x3+x2x3x4+x3x4x5+x4x5x1+x5x1x2 x2

sage: f4=x1x2x3x4+x2x3x4x5+x3x4x5x1+x4x5x1x2+x5x1x2*x3

sage: f5=x1x2x3 sage: f5=x1x2x3x4x5-1 x5-1

sage: I = Ideal(f1,f2,f3,f4,f5) Ideal(f1,f2,f3,f4,f5)

sage: I.variety()

produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation. implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]

which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level of the output but without success.

Thanks for your time!

### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1x2+x2x3+x3x4+x4x5+x1*x5

sage: f3=x1x2x3+x2x3x4+x3x4x5+x4x5x1+x5x1x2

sage: f4=x1x2x3x4+x2x3x4x5+x3x4x5x1+x4x5x1x2+x5x1x2*x3

sage: f5=x1x2x3x4x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()

(some of the asterisks appear to be showing up better than others for some reason---not sure why some seem to have disappeared. They're all there, though!)

produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]

which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level of the output but without success.

Thanks for your time!

### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1x2+x2x3+x3x4+x4x5+x1*x5

sage: f3=x1x2x3+x2x3x4+x3x4x5+x4x5x1+x5x1x2

sage: f4=x1x2x3x4+x2x3x4x5+x3x4x5x1+x4x5x1x2+x5x1x2*x3

sage: f5=x1x2x3x4x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()

(some of the multiplication asterisks appear to be showing up better than disappear in the live post---not entirely sure why, as others for some reason---not sure why some seem to have disappeared. appear just as they should. They're all there, though!)in there and the evaluates successfully on my end, though, for what it's worth! Sorry for the inconvenience.)

produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]

which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level of the output but without success.

Thanks for your time!

### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex') sage: f1=x1+x2+x3+x4+x5 sage: f2=x1x2+x2x3+x3x4+x4x5+x1*x5 sage: f3=x1x2x3+x2x3x4+x3x4x5+x4x5x1+x5x1x2 sage: f4=x1x2x3x4+x2x3x4x5+x3x4x5x1+x4x5x1x2+x5x1x2*x3 sage: f5=x1x2x3x4x5-1 PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1*x2+x2*x3+x3*x4+x4*x5+x1*x5

sage: f3=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2

sage: f4=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3

sage: f5=x1*x2*x3*x4*x5-1

sage: I = Ideal(f1,f2,f3,f4,f5) sage: I.variety() (some of the multiplication asterisks appear to disappear in the live post---not entirely sure why, as others appear just as they should. They're all in there and the evaluates successfully on my end, though, for what it's worth! Sorry for the inconvenience.)Ideal(f1,f2,f3,f4,f5)

sage: I.variety()


produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation. implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]1.00000000000000}]


which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level of the output but without success.

Thanks for your time!

### I.variety() : verbose 0, missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1*x2+x2*x3+x3*x4+x4*x5+x1*x5

sage: f3=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2

sage: f4=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3

sage: f5=x1*x2*x3*x4*x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()


produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]


which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

Why isn't Sage finding and/or reporting all of this system's solutions? Is it related to the verbose 0 (2403: multi_polynomial_ideal.py, variety) message? I've been googling for ways to alter the verbosity level perhaps placing the system into a suitable Groebner basis, solving the univariate x5 equation(s), and then giving up on the x1-x4's, but not telling us that it's doing so? Or is it just suppressing some of the output but without success.for some reason? (or.. something else entirely that I haven't thought of!)

Thanks for your time!

### I.variety() : missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1*x2+x2*x3+x3*x4+x4*x5+x1*x5

sage: f3=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2

sage: f4=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3

sage: f5=x1*x2*x3*x4*x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()


produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]


which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

If I change the order from lex to degrevlex, I.variety() fails with the following error message:

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-479-cb6d916bc8b3> in <module>()
----> 1 I.variety()

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in __call__(self, *args, **kwds)
603         if not R.base_ring().is_field():
604             raise ValueError("Coefficient ring must be a field for function '%s'."%(self.f.__name__))
--> 605         return self.f(self._instance, *args, **kwds)
606
607 require_field = RequireField

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in variety(self, ring)
2667           if self.ring().term_order().is_global():
2668             verbose("Warning: falling back to very slow toy implementation.", level=0)
-> 2669             T = toy_variety.triangular_factorization(self.groebner_basis())
2670           else:
2671             raise TypeError("Local/unknown orderings not supported by 'toy_buchberger' implementation.")

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/toy_variety.pyc in triangular_factorization(B, n)
279   # recursively build the family,
280   # looping through the factors of p
--> 281   for (q,a) in p.factor():
282     # Construct an analog to I in (R.quotient(R.ideal(q)))[x_0,x_1,...x_{n-1}]
283     I = R.ideal([each.reduce([q]) for each in G])

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_element.pyc in factor(self, proof)
1662                 raise NotImplementedError("Factorization of multivariate polynomials over prime fields with characteristic > 2^29 is not implemented.")
1663         if proof:
-> 1664             raise NotImplementedError("proof = True factorization not implemented.  Call factor with proof=False.")
1665
1666         R._singular_().set_ring()


which is kind've mystifying, since we're working over either RR or CC, both of which have characteristic 0, and neither of which is a prime field. I thought maybe this is related to I.variety()'s odd behavior under lex order somehow, though, so it'd be worth including here.

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ).

). Why isn't Sage finding and/or reporting all of this system's solutions? Is Under lex order, is it perhaps placing the system into a suitable Groebner basis, solving the univariate x5 equation(s), and then giving up on the x1-x4's, but not telling us that it's doing so? Or is it just suppressing some of the output for some reason? (or.. something else entirely that I haven't thought of!)

Thanks for your time!

### I.variety() : missing solution values

Hi there!

The following code:

sage: R.<x1,x2,x3,x4,x5> = PolynomialRing(RR,5,order='lex')

sage: f1=x1+x2+x3+x4+x5

sage: f2=x1*x2+x2*x3+x3*x4+x4*x5+x1*x5

sage: f3=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2

sage: f4=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3

sage: f5=x1*x2*x3*x4*x5-1

sage: I = Ideal(f1,f2,f3,f4,f5)

sage: I.variety()


produces

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.

[{x5: -2.61803398874989}, {x5: -0.381966011250105}, {x5: 1.00000000000000}]


which seems to be missing a large number of solution values, and only producing values for x5 even in those solutions in reports on, for some reason. A similar problem occurs if I solve over CC rather than RR (but involves a lot more text, so this seemed nicer to copy/paste).

If I change the order from lex to degrevlex, I.variety() fails with the following error message:

verbose 0 (2403: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-479-cb6d916bc8b3> in <module>()
----> 1 I.variety()

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in __call__(self, *args, **kwds)
603         if not R.base_ring().is_field():
604             raise ValueError("Coefficient ring must be a field for function '%s'."%(self.f.__name__))
--> 605         return self.f(self._instance, *args, **kwds)
606
607 require_field = RequireField

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc in variety(self, ring)
2667           if self.ring().term_order().is_global():
2668             verbose("Warning: falling back to very slow toy implementation.", level=0)
-> 2669             T = toy_variety.triangular_factorization(self.groebner_basis())
2670           else:
2671             raise TypeError("Local/unknown orderings not supported by 'toy_buchberger' implementation.")

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/toy_variety.pyc in triangular_factorization(B, n)
279   # recursively build the family,
280   # looping through the factors of p
--> 281   for (q,a) in p.factor():
282     # Construct an analog to I in (R.quotient(R.ideal(q)))[x_0,x_1,...x_{n-1}]
283     I = R.ideal([each.reduce([q]) for each in G])

/Applications/sage/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_element.pyc in factor(self, proof)
1662                 raise NotImplementedError("Factorization of multivariate polynomials over prime fields with characteristic > 2^29 is not implemented.")
1663         if proof:
-> 1664             raise NotImplementedError("proof = True factorization not implemented.  Call factor with proof=False.")
1665
1666         R._singular_().set_ring()


which is kind've mystifying, since we're working over either RR or CC, both of which have characteristic 0, and neither of which is a prime field. I thought maybe this is related to I.variety()'s odd behavior under lex order somehow, though, so it'd be worth including here.

I know this system has 10 real solutions (it's an example from HOM4PS2's documentation, and I've solved it using HOM4PS2 without issue: http://www.math.nsysu.edu.tw/~leetsung/works/HOM4PS_soft_files/HOM4PS_Linux.htm ). Why isn't Sage finding and/or reporting all of this system's solutions? Under lex order, is it perhaps placing the system into a suitable Groebner basis, solving the univariate x5 equation(s), and then giving up on the x1-x4's, but not telling us that it's doing so? Or is it just suppressing some of the output for some reason? (or.. something else entirely that I haven't thought of!)

Thanks for your time!

p.s. Using the singular_console() in Sage and adapting this example (: http://www.singular.uni-kl.de/Manual/latest/sing_887.htm#SEC938 ) to the present case seems to work perfectly, so perhaps this is just a limitation of Sage's implementation of Buchberger's algorithm & variety-solving? I'd assumed I.variety() would implement a wrapper to some appropriate Singular technique, but maybe that's not the case.