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How do I Pass a tuple as an argument for a multivariate polynomial?

asked 2010-08-29 20:57:29 -0500

allmar gravatar image

updated 2011-04-28 09:48:21 -0500

Kelvin Li gravatar image

I can't find any support documentation on this, but I'm sure it must be possible. To give some context, I'm working on a module for invariant theory which allows for computing matrices acting on polynomials:

Say I define a polynomial h(x1,x2) = a*x1^2 + b*x1x2 + c*x2^2 in QQ[x1,x2], and an ordered pair (2-tuple) v = (x1,-x2). How do I pass v such that h(v) is h(x1,-x2) ? In other words, I want to assign each coordinate of the tuple v to it's corresponding coordinate of the argument of h. Actually, a generalization to a polynomial in n variables which takes an n-tuple as an argument would be the most helpful. Below is the error that I received when trying to do this:

TypeError                                 Traceback (most recent call last)

/home/martin/Sage/sage-4.5.2/<ipython console> in <module>()

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/symbolic/ in sage.symbolic.expression.Expression.__call__ (sage/symbolic/expression.cpp:15476)()

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/symbolic/callable.pyc in _call_element_(self, _the_element, *args, **kwds)

449         d = dict(zip(map(repr, self.arguments()), args))

450         d.update(kwds)

--> 451         return SR(_the_element.substitute(**d))

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/symbolic/ in sage.symbolic.expression.Expression.substitute (sage/symbolic/expression.cpp:14850)()

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/symbolic/ in sage.symbolic.expression.Expression.coerce_in (sage/symbolic/expression.cpp:10193)()

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/structure/ in sage.structure.parent_old.Parent._coerce_ (sage/structure/parent_old.c:3288)()

/home/martin/Sage/sage-4.5.2/local/lib/python2.6/site-packages/sage/structure/ in sage.structure.parent.Parent.coerce (sage/structure/parent.c:7045)()

TypeError: no canonical coercion from Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x1, x2 over Rational Field to Callable function ring with arguments (x1, x2)
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you can put <pre> ... </pre> around the code block to make it more readable (prevent markdown formatting)

niles gravatar imageniles ( 2010-08-30 01:58:48 -0500 )edit

the same effect has selecting a block of text and clicking the editor button with zeroes and ones.

Evgeny gravatar imageEvgeny ( 2010-08-30 07:17:31 -0500 )edit

if you select a span of text and click "code formatting" button - the selection will be "backticked" and as a result displayed with a fixed width font.

Evgeny gravatar imageEvgeny ( 2010-08-30 07:18:40 -0500 )edit

1 answer

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answered 2010-08-29 21:36:36 -0500

updated 2010-08-29 21:37:17 -0500


Here is a possibility

sage: R.<a,b> = PolynomialRing(QQ,['a','b'])
sage: R
Multivariate Polynomial Ring in a, b over Rational Field
sage: P = a^3*b^5+3*a^2+2*b^4
sage: P
sage: P(5,6)   # standard way
sage: P(a=5,b=6)  # another one
sage: v = (5,6)
sage: P(*v)   # the * operator unfold a tuple as 5,6
sage: d = {'a': 5, 'b': 6}
sage: P(**v)  # the ** operator unfold a dictionary as a=5,b=6

But if v is a vector and not a tuple the trick won't work.

Hoping this would be useful, Vincent

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And for the many-variable case, you can use `d = dict(zip(R.gens(),v))` and `h.subs(d)` (I think the post above forgot to mention this last part). If v really has to be a vector, you could probably convert it to a tuple or list first.

niles gravatar imageniles ( 2010-08-30 01:54:55 -0500 )edit

Niles: for me P.subs(a=5,b=6) does not work for an element of a polynomial ring (but does for symbolic expressions).

vdelecroix gravatar imagevdelecroix ( 2010-08-30 02:26:32 -0500 )edit

Digression: flatten() is useful in related use-cases. The max_level keyword argument is very useful when there is multiple nesting.

ccanonc gravatar imageccanonc ( 2010-08-30 07:15:05 -0500 )edit

vdelecroix: I think you have to pass the dictionary object to `subs()`

niles gravatar imageniles ( 2010-08-30 09:11:43 -0500 )edit

What about the efficiency of each/any of the above methods? In particular, is unpacking the dictionary quicker than unpacking the list?

tdstephens3 gravatar imagetdstephens3 ( 2011-06-14 10:16:16 -0500 )edit

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Asked: 2010-08-29 20:57:29 -0500

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Last updated: Aug 30 '10