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 anime...@iiits.in 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 anime...@iiits.in >>> 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 anime...@iiits.in >>>>>>> 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? >>>>>>>> >>>>>>> -- You received this message because you are subscribed to the Google Groups "sage-gsoc" group. 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sage: T = TropicalSemiring(QQ) sage: PolynomialRing(T, 'x') Univariate Polynomial Ring in x over Tropical semiring over Rational Field sage: p = PolynomialRing(T, 'x,y') sage: p Multivariate Polynomial Ring in x, y over Tropical semiring over Rational Field sage: p.cardinality() +Infinity sage: sage: sage: sage: p.base() Tropical semiring over Rational Field sage: sage: sage: sage: sage: p.gens() (<repr(<sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict at 0x7f2a99244820>) failed: ArithmeticError: cannot negate any non-infinite element>, <repr(<sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict at 0x7f2a99244e10>) failed: ArithmeticError: cannot negate any non-infinite element>) sage:
