Sorry, I mistakenly attached wrong file.
This is the correct one
On Sunday, March 31, 2024 at 11:36:25 AM UTC+5:30 Animesh Shree wrote:
> Actually I was facing few errors.
> The 4th error made me feel like coercion was required.
>
>
> On Sunday, March 31, 2024 at 3:24:02 AM UTC+5:30 tcscrims wrote:
>
>> First, coercion is not in any way essential for polynomial rings. It
>> makes some things a bit more cumbersome, but it is certainly workable by
>> putting elements in the tropical semiring.
>>
>> Second, the operations between, e.g., integers and tropical ring elements
>> does *not* make sense. What is T(1) + 1? Is it 1 or 2? There's no good way
>> to answer this. The case with (integer) 0 under addition is a special case
>> that I would like to not have, but changing that would be more work than
>> the benefit.
>>
>> Best,
>> Travis
>>
>>
>> On Sunday, March 31, 2024 at 6:45:12 AM UTC+9 [email protected] wrote:
>>
>>> I read about Matrix Algebra over Tropical Semiring and found that there
>>> is "tropical determinant" for this purpose (Page 3 : Eq 4
>>> <http://www.math.uni-konstanz.de/~michalek/may22.pdf>). This
>>> formulation does not use (additive) inversion.
>>>
>>> I was also trying different operations between tropical elements and
>>> other elements(like Integers). It looks like there is a need for
>>> implementation of coercion maps between few datatypes and tropical
>>> elements.
>>> One can check this by running few commands like
>>> sage: T = TropicalSemiring(QQ)
>>> sage: a = T(1); b = T(0)
>>> sage: a + 0; b + 0
>>> sage: a * 0; b * 0
>>> sage: a + 1; b + 1
>>> sage: a * 1; b * 1
>>>
>>> Coercion maps and models are crucial for polynomial element
>>> implementation.
>>> For example : between
>>> 'Symbolic Ring' and 'Tropical semiring over Rational Field'
>>> 'Univariate Polynomial Ring in x over Tropical semiring over Rational
>>> Field' and 'Tropical semiring over Rational Field'
>>> 'Tropical semiring over Rational Field' and 'Integer Ring'
>>> On Friday, March 29, 2024 at 12:16:44 AM UTC+5:30 tcscrims wrote:
>>>
>>>> One needs to be *very* careful about what operations mean. -T(2), the
>>>> (tropical) additive inverse of 2, is *not* T(-2). There is no way to
>>>> have min(2, a) = \infty (note that this is the (tropical) additive unit).
>>>>
>>>> I am not sure how (or if) determinants over the tropical semiring are
>>>> defined, but one could define it by having the sign included in each term.
>>>> So for the 2x2 case, it could be T(a*b) + T(-b*d), but I don’t know if
>>>> this
>>>> makes det a group homomorphism.
>>>>
>>>> Best,
>>>> Travis
>>>>
>>>>
>>>> On Wednesday, March 27, 2024 at 2:17:39 AM UTC+9 [email protected]
>>>> wrote:
>>>>
>>>>> This problem can be seen clearly if we use *Matrix_generic_dense* as
>>>>> element class for matrix space.
>>>>>
>>>>>
>>>>> On Tuesday, March 26, 2024 at 9:42:45 PM UTC+5:30 Animesh Shree wrote:
>>>>>
>>>>>> I was going through this code and got error. But I could not
>>>>>> understand why this is the case.
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> sage: T = TropicalSemiring(QQ)
>>>>>> sage: T(1)
>>>>>> 1
>>>>>> sage: T(2)
>>>>>> 2
>>>>>> sage: T(-2)
>>>>>> -2
>>>>>> sage: -T(2)
>>>>>>
>>>>>> ---------------------------------------------------------------------------
>>>>>> ArithmeticError Traceback (most recent call
>>>>>> last)
>>>>>> Cell In[26], line 1
>>>>>> ----> 1 -T(Integer(2))
>>>>>>
>>>>>> File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
>>>>>> sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
>>>>>> 276 if self._val is None:
>>>>>> 277 return self
>>>>>> --> 278 raise ArithmeticError("cannot negate any non-infinite
>>>>>> element")
>>>>>> 279
>>>>>> 280 cpdef _mul_(left, right) noexcept:
>>>>>>
>>>>>> ArithmeticError: cannot negate any non-infinite element
>>>>>> sage:
>>>>>>
>>>>>>
>>>>>> It looks like starting with T(-2) and reaching to -2 from T(2) by
>>>>>> comparing with zero(+Inf) are different things.
>>>>>> T(-2) = -2
>>>>>> T(2) -T(2) = T.zero(+inf) = T(2) + (-T(2))
>>>>>>
>>>>>> My doubt is : if we cannot negate the elements, then how can we
>>>>>> compute the determinant of a Matrix over Tropical Semiring.
>>>>>> For example in 2x2 matrix : [[a,b], [c,d]] the determinant should be
>>>>>> ad - bc
>>>>>> But can it be expressed as ad + (-bc) -->> min ( add(a,d),
>>>>>> -1*add(b,c) )?
>>>>>>
>>>>>> In fact we connot even do matrix subtraction directly.
>>>>>> What can be done in these cases??
>>>>>> On Thursday, March 21, 2024 at 8:48:26 PM UTC+5:30 Animesh Shree
>>>>>> wrote:
>>>>>>
>>>>>>> Hello Sir,
>>>>>>>
>>>>>>> I have submitted my proposal.
>>>>>>> Please review it and let me know necessary updates and improvements.
>>>>>>>
>>>>>>> I want to verify soundness of my approach and extend the proposal
>>>>>>> for Multivariate Polynomials.
>>>>>>>
>>>>>>> Thank You
>>>>>>> Animesh Shree
>>>>>>> On Tuesday, March 12, 2024 at 12:26:38 AM UTC+5:30 tcscrims wrote:
>>>>>>>
>>>>>>>> Mathematically speaking, you can always weaken axioms. However,
>>>>>>>> there are some extra advantages that additive groups have that
>>>>>>>> commutative
>>>>>>>> semirings don't have (mainly 0, the additive identity).
>>>>>>>>
>>>>>>>> That being said, there isn't anything prevent you from constructing
>>>>>>>> the appropriate categories. It would be good to have a more specific
>>>>>>>> use-case in mind, but that isn't necessary. However, one should be
>>>>>>>> careful
>>>>>>>> with the name because it would conflict with what "most" people would
>>>>>>>> call
>>>>>>>> an algebra (which is why we have MagmaticAlgebras).
>>>>>>>>
>>>>>>>> Best,
>>>>>>>> Travis
>>>>>>>>
>>>>>>>>
>>>>>>>> On Tuesday, March 12, 2024 at 2:57:29 AM UTC+9 [email protected]
>>>>>>>> wrote:
>>>>>>>>
>>>>>>>>> Hello Sir,
>>>>>>>>>
>>>>>>>>> I was going through "Algebras" and I had a doubt.
>>>>>>>>> Does MagmaticAlgebra and AssociativeAlgebra have to be implemented
>>>>>>>>> over Ring only?
>>>>>>>>> I went through internet and google uses rings to define those
>>>>>>>>> algebras, but the axioms that those algebra follow (Unital,
>>>>>>>>> Associative)
>>>>>>>>> are also preserved by commutative semirings.
>>>>>>>>> Would it be good to define MagmaticAlgebra and AssociativeAlgebra
>>>>>>>>> over other objects that follow those axioms too or come-up with
>>>>>>>>> alternative
>>>>>>>>> Algebra?
>>>>>>>>>
>>>>>>>>
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sage: T = TropicalSemiring(QQ)
sage: p = PolynomialRing(T, 'x'); p.element_class
<class 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: p(x+1)
x + 1
sage: p(x+1)(T(1))
1
sage:
sage:
sage:
sage:
sage:
sage:
sage: p(x+1)(1) # -----------------------------------Error 1 of
4-----------------------------------
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[26], line 1
----> 1 p(x+Integer(1))(Integer(1))
File ~/sage/src/sage/rings/polynomial/polynomial_element.pyx:905, in
sage.rings.polynomial.polynomial_element.Polynomial.__call__()
903 # This can save lots of coercions when the common parent is the
904 # polynomial's base ring (e.g., for evaluations at integers).
--> 905 cst, aa = coercion_model.canonical_coercion(cst, a)
906 # Use fast multiplication actions like matrix × scalar.
907 # If there is no action, replace a by an element of the
File ~/sage/src/sage/structure/coerce.pyx:1424, in
sage.structure.coerce.CoercionModel.canonical_coercion()
1422 self._record_exception()
1423
-> 1424 raise TypeError("no common canonical parent for objects with
parents: '%s' and '%s'"%(xp, yp))
1425
1426 cpdef coercion_maps(self, R, S) noexcept:
TypeError: no common canonical parent for objects with parents: 'Tropical
semiring over Rational Field' and 'Integer Ring'
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage: p(x+1)(0) # -----------------------------------Error 2 of
4-----------------------------------
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call last)
Cell In[25], line 1
----> 1 p(x+Integer(1))(Integer(0))
File ~/sage/src/sage/rings/polynomial/polynomial_element.pyx:910, in
sage.rings.polynomial.polynomial_element.Polynomial.__call__()
908 # target parent.
909 S = parent(aa)
--> 910 if coercion_model.get_action(S, R) is None:
911 a = aa
912 R = S
File ~/sage/src/sage/structure/coerce.pyx:1759, in
sage.structure.coerce.CoercionModel.get_action()
1757 except KeyError:
1758 pass
-> 1759 action = self.discover_action(R, S, op, r, s)
1760 action = self.verify_action(action, R, S, op)
1761 self._action_maps.set(R, S, op, action)
File ~/sage/src/sage/structure/coerce.pyx:1900, in
sage.structure.coerce.CoercionModel.discover_action()
1898 """
1899 if isinstance(R, Parent):
-> 1900 action = (<Parent>R).get_action(S, op, True, r, s)
1901 if action is not None:
1902 return action
File ~/sage/src/sage/structure/parent.pyx:2587, in
sage.structure.parent.Parent.get_action()
2585 action = self._get_action_(S, op, self_on_left)
2586 if action is None:
-> 2587 action = self.discover_action(S, op, self_on_left, self_el, S_el)
2588
2589 if action is not None:
File ~/sage/src/sage/structure/parent.pyx:2683, in
sage.structure.parent.Parent.discover_action()
2681 from sage.structure.coerce_actions import IntegerMulAction
2682 try:
-> 2683 return IntegerMulAction(S, self, not self_on_left, self_el)
2684 except TypeError:
2685 _record_exception()
File ~/sage/src/sage/structure/coerce_actions.pyx:743, in
sage.structure.coerce_actions.IntegerMulAction.__init__()
741 if m is None:
742 m = M.an_element()
--> 743 test = m + (-m) # make sure addition and negation is allowed
744 super().__init__(Z, M, is_left, operator.mul)
745
File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
276 if self._val is None:
277 return self
--> 278 raise ArithmeticError("cannot negate any non-infinite element")
279
280 cpdef _mul_(left, right) noexcept:
ArithmeticError: cannot negate any non-infinite element
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage: p(x+1)(T(0)) # -----------------------------------Error 3 of
4-----------------------------------
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[5], line 1
----> 1 p(x+Integer(1))(T(Integer(0)))
File ~/sage/src/sage/structure/parent.pyx:901, in
sage.structure.parent.Parent.__call__()
899 if mor is not None:
900 if no_extra_args:
--> 901 return mor._call_(x)
902 else:
903 return mor._call_with_args(x, args, kwds)
File ~/sage/src/sage/structure/coerce_maps.pyx:292, in
sage.structure.coerce_maps.NamedConvertMap._call_()
290 raise TypeError(f"cannot coerce {x} to {C}")
291 cdef Map m
--> 292 cdef Element e = method(C)
293 if e is None:
294 raise RuntimeError("BUG in coercion model: {} method of {} returned
None".format(self.method_name, type(x)))
File ~/sage/src/sage/symbolic/expression.pyx:7721, in
sage.symbolic.expression.Expression._polynomial_()
7719 else:
7720 return R([self])
-> 7721 return self.polynomial(None, ring=R)
7722
7723 def fraction(self, base_ring):
File ~/sage/src/sage/symbolic/expression.pyx:7595, in
sage.symbolic.expression.Expression.polynomial()
7593 """
7594 from sage.symbolic.expression_conversions import polynomial
-> 7595 return polynomial(se lf, base_ring=base_ring, ring=ring)
7596
7597 def laurent_polynomial(self, base_ring=None, ring=None):
File ~/sage/src/sage/symbolic/expression_conversions.py:1089, in polynomial(ex,
base_ring, ring)
1030 """
1031 Return a polynomial from the symbolic expression ``ex``.
1032
(...)
1086 1.87813065119873*x^2 + 20.0855369231877
1087 """
1088 converter = PolynomialConverter(ex, base_ring=base_ring, ring=ring)
-> 1089 res = converter()
1090 return converter.ring(res)
File ~/sage/src/sage/symbolic/expression_conversions.py:212, in
Converter.__call__(self, ex)
210 div = self.get_fake_div(ex)
211 return self.arithmetic(div, div.operator())
--> 212 return self.arithmetic(ex, operator)
213 elif operator in relation_operators:
214 return self.relation(ex, operator)
File ~/sage/src/sage/symbolic/expression_conversions.py:1026, in
PolynomialConverter.arithmetic(self, ex, operator)
1024 operator = mul
1025 ops = [self(a) for a in ex.operands()]
-> 1026 return reduce(operator, ops)
File ~/sage/src/sage/structure/element.pyx:1221, in
sage.structure.element.Element.__add__()
1219 # Left and right are Sage elements => use coercion model
1220 if BOTH_ARE_ELEMENT(cl):
-> 1221 return coercion_model.bin_op(left, right, add)
1222
1223 cdef long value
File ~/sage/src/sage/structure/coerce.pyx:1278, in
sage.structure.coerce.CoercionModel.bin_op()
1276 # We should really include the underlying error.
1277 # This causes so much headache.
-> 1278 raise bin_op_exception(op, x, y)
1279
1280 cpdef canonical_coercion(self, x, y) noexcept:
TypeError: unsupported operand parent(s) for +: 'Univariate Polynomial Ring in
x over Tropical semiring over Rational Field' and 'Tropical semiring over
Rational Field'
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage: z = T(0)
sage: p(x+T(1))(z) # -----------------------------------Error 4 of
4-----------------------------------
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[5], line 1
----> 1 p(x+T(Integer(1)))(z)
File ~/sage/src/sage/structure/element.pyx:1221, in
sage.structure.element.Element.__add__()
1219 # Left and right are Sage elements => use coercion model
1220 if BOTH_ARE_ELEMENT(cl):
-> 1221 return coercion_model.bin_op(left, right, add)
1222
1223 cdef long value
File ~/sage/src/sage/structure/coerce.pyx:1278, in
sage.structure.coerce.CoercionModel.bin_op()
1276 # We should really include the underlying error.
1277 # This causes so much headache.
-> 1278 raise bin_op_exception(op, x, y)
1279
1280 cpdef canonical_coercion(self, x, y) noexcept:
TypeError: unsupported operand parent(s) for +: 'Symbolic Ring' and 'Tropical
semiring over Rational Field'
sage:
