gh-38291: Implement a custom class for tropical polynomials
    
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This pull request introduces some new class for handling tropical
polynomials and its related functionality.

### Summary of changes:

1. Added a new class: `TropicalPolynomialSemiring` as a parent of
univariate tropical polynomials and incorporated this class into the
`PolynomialRing` constructor.
2. Added a new class: `TropicalPolynomial` as an element of univariate
tropical polynomials, which include methods like:

- `roots`: return the list of tropical roots, counted with multiplicity
- `splif_form`: return the tropical polynomial which has the same roots,
but which can be reduced to its linear factors
- `factor`: perform factorization
- `piecewise_function`: return the tropical polynomial function
- `plot`: plotting the tropical polynomial function

3. Added a new class: `TropicalMPolynomialSemiring` as a parent of
multivariate tropical polynomials and incorporated this class into the
`PolynomialRing` constructor.
4. Added a new class: `TropicalMPolynomial` as an element of
multivariate tropical polynomials, which include methods like:

- `plot3d`: return the 3d plot of tropical polynomials in two variables
- `tropical_variety`: find tropical variety for any multivariate
tropical polynomials

5. Added a new class: `TropicalVariety` and its subclass `TropicalCurve`
and `TropicalSurface` to handle the result of `tropical_variety()` of
multivariate tropical polynomials. The `TropicalVariety` include methods
like:

- `__init__`: calculation of tropical variety
- `dimension`: return the dimension of tropical variety
- `number_of_components`: return the number of components that make up
this tropical variety
- `components`: show all the components of this tropical variety
- `_components_intersection`: return the intersection of three or more
components

6. Implement the `plot` method inside the two subclasses,
`TropicalCurve` and `TropicalSurface`.

Fixes #37962.

CC: @tscrim

### 📝 Checklist

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- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: #38291
Reported by: Verrel Rievaldo Wijaya
Reviewer(s): Travis Scrimshaw, Verrel Rievaldo Wijaya

Author: Release Manager

src/doc/en/reference/polynomial_rings/index.rst

@@ -62,6 +62,16 @@ Infinite Polynomial Rings
    sage/rings/polynomial/symmetric_ideal
    sage/rings/polynomial/symmetric_reduction
 
+Tropical Polynomials
+--------------------
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/rings/semirings/tropical_polynomial
+   sage/rings/semirings/tropical_mpolynomial
+   sage/rings/semirings/tropical_variety
+
 Boolean Polynomials
 -------------------
 

src/doc/en/reference/references/index.rst

@@ -857,6 +857,9 @@ REFERENCES:
 .. [BrHu2019] Petter Brändén, June Huh. *Lorentzian polynomials*.
               Ann. Math. (2) 192, No. 3, 821-891 (2020).
               :arxiv:`1902.03719`, :doi:`10.4007/annals.2020.192.3.4`.
+        
+.. [Bru2014] Erwan Brugalle and Kristin Shaw. *A bit of tropical geometry*.
+             Amer. Math. Monthly, 121(7):563-589, 2014.
 
 .. [BHNR2004] \S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the
               Palindromic Complexity of Infinite Words,
@@ -2612,6 +2615,10 @@ REFERENCES:
             Wehler K3 Surfaces over finite fields*. New Zealand Journal
             of Mathematics 45 (2015), 19–31.
 
+.. [Fil2017] Ivana Filipan, *An Invitation to Combinatorial Tropical Geometry*.
+             Conference: 1st Croatian Combinatorial Days. 2017.
+             :doi:`10.5592/CO/CCD.2016.05`.
+
 .. [FIV2012] \H. Fournier, A. Ismail, and A. Vigneron. *Computing the Gromov
              hyperbolicity of a discrete metric space*. 2012.
              :arxiv:`1210.3323`.

src/sage/rings/polynomial/multi_polynomial.pyx

@@ -2665,7 +2665,7 @@ cdef class MPolynomial(CommutativePolynomial):
 
         if gens:
             # substituting all variables (in a polynomial ring with variables) with 0
-            d = {str(gen): 0 for gen in gens}
+            d = {str(gen): self.base_ring().zero() for gen in gens}
             tester.assertEqual(self.subs(**d).parent(), self.parent().base_ring())
 
             # substituting all variables (in a polynomial ring with variables)
@@ -2678,7 +2678,7 @@ cdef class MPolynomial(CommutativePolynomial):
 
         if len(gens) > 1:
             # substituting one variable (in a polynomial ring with variables) with 0
-            d = {str(gens[0]): 0}
+            d = {str(gens[0]): self.base_ring().zero()}
             tester.assertEqual(self.subs(**d).parent(), self.parent())
 
             # test error checking: partial substitution by elements

src/sage/rings/polynomial/multi_polynomial_element.py

@@ -176,9 +176,14 @@ def __call__(self, *x, **kwds):
             K = x[0].parent()
         except AttributeError:
             K = self.parent().base_ring()
-        y = K(0)
+        try:
+            y = K.zero()
+            one = K.one()
+        except (AttributeError, RuntimeError):
+            y = K(0)
+            one = K(1)
         for m, c in self.element().dict().items():
-            y += c * prod(v ** e for v, e in zip(x, m) if e)
+            y += c * prod((v ** e for v, e in zip(x, m) if e), one)
         return y
 
     def _richcmp_(self, right, op):

src/sage/rings/polynomial/polydict.pyx

@@ -800,7 +800,9 @@ cdef class PolyDict:
             ring = self.__repn[E[0]].parent()
             pos_one = ring.one()
             neg_one = -pos_one
-        except AttributeError:
+        except (AttributeError, ArithmeticError):
+            # AritchmeticError occurs when self.__repn[E[0]] is a tropical
+            # semiring element
             # probably self.__repn[E[0]] is not a ring element
             pos_one = 1
             neg_one = -1
@@ -901,7 +903,9 @@ cdef class PolyDict:
             ring = self.__repn[E[0]].parent()
             pos_one = ring.one()
             neg_one = -pos_one
-        except AttributeError:
+        except (AttributeError, ArithmeticError):
+            # AritchmeticError occurs when self.__repn[E[0]] is a tropical
+            # semiring element
             # probably self.__repn[E[0]] is not a ring element
             pos_one = 1
             neg_one = -1

src/sage/rings/polynomial/polynomial_element.pyx

@@ -6051,7 +6051,7 @@ cdef class Polynomial(CommutativePolynomial):
 
         - Naqi Jaffery (2006-01-24): examples
         """
-        return not self.is_zero() and self[self.degree()] == 1
+        return not self.is_zero() and self[self.degree()] == self.base_ring().one()
 
     def is_unit(self):
         r"""

src/sage/rings/polynomial/polynomial_ring_constructor.py

@@ -537,7 +537,7 @@ def PolynomialRing(base_ring, *args, **kwds):
         sage: PolynomialRing(4)
         Traceback (most recent call last):
         ...
-        TypeError: base_ring 4 must be a ring
+        TypeError: base_ring 4 must be a ring or the tropical semiring
         sage: PolynomialRing(QQ, -1)
         Traceback (most recent call last):
         ...
@@ -622,8 +622,9 @@ def PolynomialRing(base_ring, *args, **kwds):
         sage: R.<x,y> = PolynomialRing(RIF,2)
         sage: TestSuite(R).run(skip=['_test_elements', '_test_elements_eq_transitive'])
     """
-    if base_ring not in Rings():
-        raise TypeError("base_ring {!r} must be a ring".format(base_ring))
+    from sage.rings.semirings.tropical_semiring import TropicalSemiring
+    if base_ring not in Rings() and not isinstance(base_ring, TropicalSemiring):
+        raise TypeError("base_ring {!r} must be a ring or the tropical semiring".format(base_ring))
 
     n = -1  # Unknown number of variables
     names = None  # Unknown variable names
@@ -792,7 +793,11 @@ def _single_variate(base_ring, name, sparse=None, implementation=None, order=Non
 
     # Generic implementations
     if constructor is None:
-        if base_ring not in _CommutativeRings:
+        from sage.rings.semirings.tropical_semiring import TropicalSemiring
+        if isinstance(base_ring, TropicalSemiring):
+            from sage.rings.semirings.tropical_polynomial import TropicalPolynomialSemiring
+            constructor = TropicalPolynomialSemiring
+        elif base_ring not in _CommutativeRings:
             constructor = polynomial_ring.PolynomialRing_general
         elif base_ring in _CompleteDiscreteValuationRings:
             constructor = polynomial_ring.PolynomialRing_cdvr
@@ -804,6 +809,7 @@ def _single_variate(base_ring, name, sparse=None, implementation=None, order=Non
             constructor = polynomial_ring.PolynomialRing_integral_domain
         else:
             constructor = polynomial_ring.PolynomialRing_commutative
+
         implementation_names = constructor._implementation_names(implementation, base_ring, sparse)
 
         # Only use names which are not supported by the specialized class.
@@ -861,7 +867,11 @@ def _multi_variate(base_ring, names, sparse=None, order='degrevlex', implementat
 
     if R is None and implementation == "generic":
         from . import multi_polynomial_ring
-        if base_ring in _Domains:
+        from sage.rings.semirings.tropical_semiring import TropicalSemiring
+        if isinstance(base_ring, TropicalSemiring):
+            from sage.rings.semirings.tropical_mpolynomial import TropicalMPolynomialSemiring
+            constructor = TropicalMPolynomialSemiring
+        elif base_ring in _Domains:
             constructor = multi_polynomial_ring.MPolynomialRing_polydict_domain
         else:
             constructor = multi_polynomial_ring.MPolynomialRing_polydict
@@ -1024,4 +1034,4 @@ def BooleanPolynomialRing_constructor(n=None, names=None, order='lex'):
 
 ############################################################################
 # END (Factory function for making polynomial rings)
-############################################################################
+############################################################################