Univariate Tropical Polynomials

AUTHORS:

  • Verrel Rievaldo Wijaya (2024-06): initial version

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x> = PolynomialRing(T)
sage: x.parent()
Univariate Tropical Polynomial Semiring in x over Rational Field
sage: (x + R(3)*x) * (x^2 + x)
3*x^3 + 3*x^2
sage: (x^2 + R(1)*x + R(-1))^2
0*x^4 + 1*x^3 + 2*x^2 + 0*x + (-2)
sage: (x^2 + x + R(0))^4
0*x^8 + 0*x^7 + 0*x^6 + 0*x^5 + 0*x^4 + 0*x^3 + 0*x^2 + 0*x + 0

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> x.parent()
Univariate Tropical Polynomial Semiring in x over Rational Field
>>> (x + R(Integer(3))*x) * (x**Integer(2) + x)
3*x^3 + 3*x^2
>>> (x**Integer(2) + R(Integer(1))*x + R(-Integer(1)))**Integer(2)
0*x^4 + 1*x^3 + 2*x^2 + 0*x + (-2)
>>> (x**Integer(2) + x + R(Integer(0)))**Integer(4)
0*x^8 + 0*x^7 + 0*x^6 + 0*x^5 + 0*x^4 + 0*x^3 + 0*x^2 + 0*x + 0

REFERENCES:

class sage.rings.semirings.tropical_polynomial.TropicalPolynomial(parent, x=None, check=True, is_gen=False, construct=False)[source]

Bases: Polynomial_generic_sparse

A univariate tropical polynomial.

A tropical polynomial is a polynomial whose coefficients come from a tropical semiring. Tropical polynomial induces a function which is piecewise linear, where each segment of the function has a non-negative integer slope. Tropical roots (zeros) of polynomial \(P(x)\) is defined as all points \(x_0\) for which the graph of \(P(x)\) change its slope. The difference in the slopes of the two pieces adjacent to this root gives the order of the root.

The tropical polynomials are implemented with a sparse format by using a dict whose keys are the exponent and values the corresponding coefficients. Each coefficient is a tropical number.

EXAMPLES:

We construct a tropical polynomial semiring by defining a base tropical semiring and then inputting it to PolynomialRing:

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x> = PolynomialRing(T); R
Univariate Tropical Polynomial Semiring in x over Rational Field
sage: R.0
0*x

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1); R
Univariate Tropical Polynomial Semiring in x over Rational Field
>>> R.gen(0)
0*x

One way to construct an element is to provide a list or tuple of coefficients. Another way to define an element is to write a polynomial equation with each coefficient converted to the semiring:

sage: p1 = R([1,4,None,0]); p1
0*x^3 + 4*x + 1
sage: p2 = R(1)*x^2 + R(2)*x + R(3); p2
1*x^2 + 2*x + 3

>>> from sage.all import *
>>> p1 = R([Integer(1),Integer(4),None,Integer(0)]); p1
0*x^3 + 4*x + 1
>>> p2 = R(Integer(1))*x**Integer(2) + R(Integer(2))*x + R(Integer(3)); p2
1*x^2 + 2*x + 3

We can do some basic arithmetic operations for these tropical polynomials. Remember that numbers given have to be in the tropical semiring. If not, then it will raise an error:

sage: p1 + p2
0*x^3 + 1*x^2 + 4*x + 3
sage: p1 * p2
1*x^5 + 2*x^4 + 5*x^3 + 6*x^2 + 7*x + 4
sage: 2 * p1
Traceback (most recent call last):
...
ArithmeticError: cannot negate any non-infinite element
sage: T(2) * p1
2*x^3 + 6*x + 3
sage: p1(3)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'Tropical semiring over Rational Field' and 'Integer Ring'
sage: p1(T(3))
9

>>> from sage.all import *
>>> p1 + p2
0*x^3 + 1*x^2 + 4*x + 3
>>> p1 * p2
1*x^5 + 2*x^4 + 5*x^3 + 6*x^2 + 7*x + 4
>>> Integer(2) * p1
Traceback (most recent call last):
...
ArithmeticError: cannot negate any non-infinite element
>>> T(Integer(2)) * p1
2*x^3 + 6*x + 3
>>> p1(Integer(3))
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'Tropical semiring over Rational Field' and 'Integer Ring'
>>> p1(T(Integer(3)))
9

We can find all tropical roots of a tropical polynomial, counted with their multiplicities:

sage: p1.roots()
[-3, 2, 2]
sage: p2.roots()
[1, 1]

>>> from sage.all import *
>>> p1.roots()
[-3, 2, 2]
>>> p2.roots()
[1, 1]

Even though every tropical polynomials have tropical roots, this does not neccessarily means it can be factored into its linear factors:

sage: p1.factor()
(0) * (0*x^3 + 4*x + 1)
sage: p2.factor()
(1) * (0*x + 1)^2

>>> from sage.all import *
>>> p1.factor()
(0) * (0*x^3 + 4*x + 1)
>>> p2.factor()
(1) * (0*x + 1)^2

Every tropical polynomial \(p(x)\) have a corresponding unique tropical polynomial \(\bar{p}(x)\) with the same roots that can be factored. We call \(\bar{p}(x)\) the tropical polynomial split form of \(p(x)\):

sage: p1.split_form()
0*x^3 + 2*x^2 + 4*x + 1
sage: p2.split_form()
1*x^2 + 2*x + 3

>>> from sage.all import *
>>> p1.split_form()
0*x^3 + 2*x^2 + 4*x + 1
>>> p2.split_form()
1*x^2 + 2*x + 3

Every tropical polynomial induce a piecewise linear function that can be invoked in the following way:

sage: p1.piecewise_function()
piecewise(x|-->1 on (-oo, -3], x|-->x + 4 on (-3, 2), x|-->3*x on [2, +oo); x)
sage: p1.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> p1.piecewise_function()
piecewise(x|-->1 on (-oo, -3], x|-->x + 4 on (-3, 2), x|-->3*x on [2, +oo); x)
>>> p1.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-1.svg 
sage: p2.piecewise_function()
piecewise(x|-->3 on (-oo, 1], x|-->2*x + 1 on (1, +oo); x)
sage: p2.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> p2.piecewise_function()
piecewise(x|-->3 on (-oo, 1], x|-->2*x + 1 on (1, +oo); x)
>>> p2.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-2.svg
factor()[source]

Return the factorization of self into its tropical linear factors.

Note that the factor \(x - x_0\) in classical algebra gets transformed to the factor \(x + x_0\), since the root of the tropical polynomial \(x + x_0\) is \(x_0\) and not \(-x_0\). However, not every tropical polynomial can be factored.

OUTPUT: :class:’Factorization’

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([6,3,1,0]); p1
0*x^3 + 1*x^2 + 3*x + 6
sage: factor(p1)
(0) * (0*x + 1) * (0*x + 2) * (0*x + 3)

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(6),Integer(3),Integer(1),Integer(0)]); p1
0*x^3 + 1*x^2 + 3*x + 6
>>> factor(p1)
(0) * (0*x + 1) * (0*x + 2) * (0*x + 3)

Such factorization is not always possible:

sage: p2 = R([4,4,2]); p2
2*x^2 + 4*x + 4
sage: p2.factor()
(2) * (0*x^2 + 2*x + 2)

>>> from sage.all import *
>>> p2 = R([Integer(4),Integer(4),Integer(2)]); p2
2*x^2 + 4*x + 4
>>> p2.factor()
(2) * (0*x^2 + 2*x + 2)
piecewise_function()[source]

Return the tropical polynomial function of self.

The function is a piecewise linear function with the domains are divided by the roots. First we convert each term of polynomial to its corresponding linear function. Next, we must determine which term achieves the minimum (maximum) at each interval.

OUTPUT: a piecewise function

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([4,2,1,3]); p1
3*x^3 + 1*x^2 + 2*x + 4
sage: p1.piecewise_function()
piecewise(x|-->4 on (-oo, 1/3], x|-->3*x + 3 on (1/3, +oo); x)

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(4),Integer(2),Integer(1),Integer(3)]); p1
3*x^3 + 1*x^2 + 2*x + 4
>>> p1.piecewise_function()
piecewise(x|-->4 on (-oo, 1/3], x|-->3*x + 3 on (1/3, +oo); x)

A constant tropical polynomial will result in a constant function:

sage: p2 = R(3)
sage: p2.piecewise_function()
3

>>> from sage.all import *
>>> p2 = R(Integer(3))
>>> p2.piecewise_function()
3

A monomial will resulted in a linear function:

sage: p3 = R(1)*x^3
sage: p3.piecewise_function()
3*x + 1

>>> from sage.all import *
>>> p3 = R(Integer(1))*x**Integer(3)
>>> p3.piecewise_function()
3*x + 1
plot(xmin=None, xmax=None)[source]

Return the plot of self, which is the tropical polynomial function we get from self.piecewise_function().

INPUT:

  • xmin – (optional) real number

  • xmax – (optional) real number

OUTPUT:

If xmin and xmax is given, then it return a plot of piecewise linear function of self with the axes start from xmin to xmax. Otherwise, the domain will start from the the minimum root of self minus 1 to maximum root of self plus 1. If the function of self is constant or linear, then the default domain will be \([-1,1]\).

EXAMPLES:

If the tropical semiring use a max-plus algebra, then the graph will be of piecewise linear convex function:

sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([4,2,1,3]); p1
3*x^3 + 1*x^2 + 2*x + 4
sage: p1.roots()
[1/3, 1/3, 1/3]
sage: p1.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(4),Integer(2),Integer(1),Integer(3)]); p1
3*x^3 + 1*x^2 + 2*x + 4
>>> p1.roots()
[1/3, 1/3, 1/3]
>>> p1.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-3.svg

A different result will be obtained if the tropical semiring employs a min-plus algebra. Rather, a graph of the piecewise linear concave function will be obtained:

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([4,2,1,3])
sage: p1.roots()
[-2, 1, 2]
sage: p1.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(4),Integer(2),Integer(1),Integer(3)])
>>> p1.roots()
[-2, 1, 2]
>>> p1.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-4.svg
roots()[source]

Return the list of all tropical roots of self, counted with multiplicity.

OUTPUT: a list of tropical numbers

ALGORITHM:

For each pair of monomials in the polynomial, we find the point where their values are equal. This is the same as solving the equation \(c_1 + a_1 \cdot x = c_2 + a_2 \cdot x\) for \(x\), where \((c_1, a_1)\) and \((c_2, a_2)\) are the coefficients and exponents of monomials.

The solution to this equation is \(x = (c_2-c_1)/(a_1-a_2)\). We substitute this \(x\) to each monomials in polynomial and check if the maximum (minimum) is achieved by the previous two monomials. If it is, then \(x\) is the root of tropical polynomial. In this case, the order of the root at \(x\) is the maximum of \(|i-j|\) for all possible pairs \(i,j\) which realise this maximum (minimum).

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([5,4,1,0,2,4,3]); p1
3*x^6 + 4*x^5 + 2*x^4 + 0*x^3 + 1*x^2 + 4*x + 5
sage: p1.roots()
[-1, -1, -1, 1, 2, 2]

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(5),Integer(4),Integer(1),Integer(0),Integer(2),Integer(4),Integer(3)]); p1
3*x^6 + 4*x^5 + 2*x^4 + 0*x^3 + 1*x^2 + 4*x + 5
>>> p1.roots()
[-1, -1, -1, 1, 2, 2]

There will be no tropical root for constant polynomial. Additionaly, for a monomial, the tropical root is assumed to be the additive identity of its base tropical semiring:

sage: p2 = R(2)
sage: p2.roots()
[]
sage: p3 = x^3
sage: p3.roots()
[+infinity, +infinity, +infinity]

>>> from sage.all import *
>>> p2 = R(Integer(2))
>>> p2.roots()
[]
>>> p3 = x**Integer(3)
>>> p3.roots()
[+infinity, +infinity, +infinity]
split_form()[source]

Return the tropical polynomial which has the same roots as self but which can be reduced to its linear factors.

If a tropical polynomial has roots at \(x_1, x_2, \ldots, x_n\), then its split form is the tropical product of linear terms of the form \((x + x_i)\) for all \(i=1,2,\ldots,n\).

OUTPUT: TropicalPolynomial

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([5,4,1,0,2,4,3]); p1
3*x^6 + 4*x^5 + 2*x^4 + 0*x^3 + 1*x^2 + 4*x + 5
sage: p1.split_form()
3*x^6 + 2*x^5 + 1*x^4 + 0*x^3 + 1*x^2 + 3*x + 5

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(5),Integer(4),Integer(1),Integer(0),Integer(2),Integer(4),Integer(3)]); p1
3*x^6 + 4*x^5 + 2*x^4 + 0*x^3 + 1*x^2 + 4*x + 5
>>> p1.split_form()
3*x^6 + 2*x^5 + 1*x^4 + 0*x^3 + 1*x^2 + 3*x + 5


sage: T = TropicalSemiring(QQ, use_min=False)
sage: R.<x> = PolynomialRing(T)
sage: p1 = R([5,4,1,0,2,4,3])
sage: p1.split_form()
3*x^6 + 4*x^5 + 21/5*x^4 + 22/5*x^3 + 23/5*x^2 + 24/5*x + 5

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> p1 = R([Integer(5),Integer(4),Integer(1),Integer(0),Integer(2),Integer(4),Integer(3)])
>>> p1.split_form()
3*x^6 + 4*x^5 + 21/5*x^4 + 22/5*x^3 + 23/5*x^2 + 24/5*x + 5
class sage.rings.semirings.tropical_polynomial.TropicalPolynomialSemiring(base_semiring, names)[source]

Bases: UniqueRepresentation, Parent

The semiring of univariate tropical polynomials.

This is the commutative semiring consisting of finite linear combinations of tropical monomials under (tropical) addition and multiplication with the identity element being \(+\infty\) (or \(-\infty\) depending on whether the base tropical semiring is using min-plus or max-plus algebra).

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R.<x> = PolynomialRing(T)
sage: f = T(1)*x
sage: g = T(2)*x
sage: f + g
1*x
sage: f * g
3*x^2
sage: f + R.zero() == f
True
sage: f * R.zero() == R.zero()
True
sage: f * R.one() == f
True

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('x',)); (x,) = R._first_ngens(1)
>>> f = T(Integer(1))*x
>>> g = T(Integer(2))*x
>>> f + g
1*x
>>> f * g
3*x^2
>>> f + R.zero() == f
True
>>> f * R.zero() == R.zero()
True
>>> f * R.one() == f
True
Element[source]

alias of TropicalPolynomial

gen(n=0)[source]

Return the indeterminate generator of self.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R.<abc> = PolynomialRing(T)
sage: R.gen()
0*abc

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('abc',)); (abc,) = R._first_ngens(1)
>>> R.gen()
0*abc
gens()[source]

Return a tuple whose entries are the generators for self.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R.<abc> = PolynomialRing(T)
sage: R.gens()
(0*abc,)

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, names=('abc',)); (abc,) = R._first_ngens(1)
>>> R.gens()
(0*abc,)
interpolation(points)[source]

Return a tropical polynomial with its function is a linear interpolation of point in points if possible.

If there is only one point, then it will give a constant polynomial. Because we are using linear interpolation, each point is actually a root of the resulted tropical polynomial.

INPUT:

  • points – a list of tuples (x, y)

OUTPUT: TropicalPolynomial

EXAMPLES:

sage: T = TropicalSemiring(QQ, use_min=True)
sage: R = PolynomialRing(T, 'x')
sage: points = [(-2,-3),(1,3),(2,4)]
sage: p1 = R.interpolation(points); p1
1*x^2 + 2*x + 4
sage: p1.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=True)
>>> R = PolynomialRing(T, 'x')
>>> points = [(-Integer(2),-Integer(3)),(Integer(1),Integer(3)),(Integer(2),Integer(4))]
>>> p1 = R.interpolation(points); p1
1*x^2 + 2*x + 4
>>> p1.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-5.svg 
sage: T = TropicalSemiring(QQ, use_min=False)
sage: R = PolynomialRing(T,'x')
sage: points = [(0,0),(1,1),(2,4)]
sage: p1 = R.interpolation(points); p1
(-2)*x^3 + (-1)*x^2 + 0*x + 0
sage: p1.plot()
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> T = TropicalSemiring(QQ, use_min=False)
>>> R = PolynomialRing(T,'x')
>>> points = [(Integer(0),Integer(0)),(Integer(1),Integer(1)),(Integer(2),Integer(4))]
>>> p1 = R.interpolation(points); p1
(-2)*x^3 + (-1)*x^2 + 0*x + 0
>>> p1.plot()
Graphics object consisting of 1 graphics primitive
../../../_images/tropical_polynomial-6.svg
is_sparse()[source]

Return True to indicate that the objects are sparse polynomials.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R = PolynomialRing(T, 'x')
sage: R.is_sparse()
True

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, 'x')
>>> R.is_sparse()
True
ngens()[source]

Return the number of generators of self, which is 1 since it is a univariate polynomial ring.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R = PolynomialRing(T, 'x')
sage: R.ngens()
1

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, 'x')
>>> R.ngens()
1
one()[source]

Return the multiplicative identity of self.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R = PolynomialRing(T, 'x')
sage: R.one()
0

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, 'x')
>>> R.one()
0
random_element(degree=(-1, 2), monic=False, *args, **kwds)[source]

Return a random tropical polynomial of given degrees (bounds).

OUTPUT: TropicalPolynomial

See also

sage.rings.polynomial.polynomial_ring.PolynomialRing_general.random_element()

EXAMPLES:

Tropical polynomial over an integer with each coefficient bounded between x and y:

sage: T = TropicalSemiring(ZZ)
sage: R = PolynomialRing(T, 'x')
sage: f = R.random_element(8, x=3, y=10)
sage: f.degree()
8
sage: f.parent() is R
True
sage: all(a >= T(3) for a in f.coefficients())
True
sage: all(a < T(10) for a in f.coefficients())
True

>>> from sage.all import *
>>> T = TropicalSemiring(ZZ)
>>> R = PolynomialRing(T, 'x')
>>> f = R.random_element(Integer(8), x=Integer(3), y=Integer(10))
>>> f.degree()
8
>>> f.parent() is R
True
>>> all(a >= T(Integer(3)) for a in f.coefficients())
True
>>> all(a < T(Integer(10)) for a in f.coefficients())
True

If a tuple of two integers is provided for the degree argument, a polynomial is selected with degrees within that range:

sage: all(R.random_element(degree=(1, 5)).degree() in range(1, 6) for _ in range(10^3))
True

>>> from sage.all import *
>>> all(R.random_element(degree=(Integer(1), Integer(5))).degree() in range(Integer(1), Integer(6)) for _ in range(Integer(10)**Integer(3)))
True

Note that the zero polynomial (\(\pm \infty\)) has degree \(-1\). To include it, set the minimum degree to \(-1\):

sage: while R.random_element(degree=(-1,2), x=-1, y=1) != R.zero():
....:     pass

>>> from sage.all import *
>>> while R.random_element(degree=(-Integer(1),Integer(2)), x=-Integer(1), y=Integer(1)) != R.zero():
...     pass

Monic polynomials are chosen among all monic polynomials with degree between the given degree argument:

sage: all(R.random_element(degree=(-1, 2), monic=True).is_monic() for _ in range(10^3))
True

>>> from sage.all import *
>>> all(R.random_element(degree=(-Integer(1), Integer(2)), monic=True).is_monic() for _ in range(Integer(10)**Integer(3)))
True
zero()[source]

Return the additive identity of self.

EXAMPLES:

sage: T = TropicalSemiring(QQ)
sage: R = PolynomialRing(T, 'x')
sage: R.zero()
+infinity

>>> from sage.all import *
>>> T = TropicalSemiring(QQ)
>>> R = PolynomialRing(T, 'x')
>>> R.zero()
+infinity