Quantum-valued polynomial rings¶

This provides a $q$ -analogue of the IntegerValuedPolynomialRing.

AUTHORS:

Frédéric Chapoton (2024-03): Initial version

class sage.rings.polynomial.q_integer_valued_polynomials.QuantumValuedPolynomialRing(R, q)[source]¶

Bases: UniqueRepresentation, Parent

The quantum-valued polynomial ring over a base ring.

Quantum-valued polynomial rings are commutative and associative algebras, with a basis indexed by nonnegative integers.

The elements are polynomials in one variable $x$ with coefficients in the field of rational functions in $q$ , such that the value at every nonegative $q$ -integer is a polynomial in $q$ .

This algebra is endowed with two bases, named B or Binomial and S or Shifted.

INPUT:

R – commutative ring
q – optional variable

There are two possible input formats:

If the argument q is not given, then the ring R is taken as a base ring and the ring of Laurent polynomials in $q$ over R is built and used.
If the argument q is given, then it should belong to the ring R and be invertible in this ring.

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(QQ).S(); F
Quantum-Valued Polynomial Ring over Rational Field
in the shifted basis

sage: F.gen()
S[1]

sage: S = QuantumValuedPolynomialRing(ZZ); S
Quantum-Valued Polynomial Ring over Integer Ring
sage: S.base_ring()
Univariate Laurent Polynomial Ring in q over Integer Ring

Quantum-valued polynomial rings commute with their base ring:

sage: K = QuantumValuedPolynomialRing(QQ).S()
sage: a = K.gen()
sage: c = K.monomial(2)

Quantum-valued polynomial rings are commutative:

sage: c^3 * a == c * a * c * c
True

We can also manipulate elements in the basis and coerce elements from our base field:

sage: F = QuantumValuedPolynomialRing(QQ).S()
sage: B = F.basis()
sage: B[2] * B[3]
(q^-5+q^-4+q^-3)*S[3] - (q^-6+2*q^-5+3*q^-4+3*q^-3+2*q^-2+q^-1)*S[4]
+ (q^-6+q^-5+2*q^-4+2*q^-3+2*q^-2+q^-1+1)*S[5]
sage: 1 - B[2] * B[2] / 2
S[0] - (1/2*q^-3)*S[2] + (1/2*q^-4+q^-3+q^-2+1/2*q^-1)*S[3]
- (1/2*q^-4+1/2*q^-3+q^-2+1/2*q^-1+1/2)*S[4]

B[source]¶: alias of Binomial

class Bases(parent_with_realization)[source]¶

Bases: Category_realization_of_parent

class ElementMethods[source]¶

Bases: object

polynomial()[source]¶

Convert to a polynomial in $x$ .

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: S = F.gen()
sage: (S+1).polynomial()
q*x + 2

sage: F = QuantumValuedPolynomialRing(ZZ).B()
sage: B = F.gen()
sage: (B+1).polynomial()
x + 1

shift(j=1)[source]¶

Shift all indices by $j$ .

INPUT:

$j$ – integer (default 1)

In the binomial basis, the shift by 1 corresponds to a summation operator from $0$ to $x$ .

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).shift()
S[1] + S[2]

sum_of_coefficients()[source]¶

Return the sum of coefficients.

In the shifted basis, this is the evaluation at $x = 0$ .

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: B = F.basis()
sage: (B[2]*B[4]).sum_of_coefficients()
1

class ParentMethods[source]¶

Bases: object

algebra_generators()[source]¶

Return the generators of this algebra.

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S(); A
Quantum-Valued Polynomial Ring over Integer Ring
in the shifted basis
sage: A.algebra_generators()
Family (S[1],)

degree_on_basis(m)[source]¶

Return the degree of the basis element indexed by m.

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: A.degree_on_basis(4)
4

from_polynomial(p)[source]¶

Convert a polynomial into the ring of quantum-valued polynomials.

This raises a ValueError if this is not possible.

INPUT:

p – a polynomial in x with coefficients in QQ(q)

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S()
sage: S = A.basis()
sage: A.from_polynomial((S[1]).polynomial())
S[1]
sage: A.from_polynomial((S[2]+2*S[3]).polynomial())
S[2] + 2*S[3]

sage: A = QuantumValuedPolynomialRing(ZZ).B()
sage: B = A.basis()
sage: A.from_polynomial((B[1]).polynomial())
B[1]
sage: A.from_polynomial((B[2]+2*B[3]).polynomial())
B[2] + 2*B[3]

gen(i=0)[source]¶

Return the generator of the algebra.

The optional argument is ignored.

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: F.gen()
S[1]

gens()[source]¶: alias of algebra_generators().

ground_ring()[source]¶

Return the ring of coefficients.

This ring is not supposed to contain the variable $q$ .

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: A.ground_ring()
Rational Field

one_basis()[source]¶

Return the number 0, which index the unit of this algebra.

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: A.one_basis()
0
sage: A.one()
S[0]

q()[source]¶

Return the variable $q$ .

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: A.q()
q

super_categories()[source]¶

Return the super-categories of self.

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ); A
Quantum-Valued Polynomial Ring over Rational Field
sage: C = A.Bases(); C
Category of bases of Quantum-Valued Polynomial Ring
over Rational Field
sage: C.super_categories()
[Category of realizations of Quantum-Valued Polynomial Ring
 over Rational Field,
 Join of Category of algebras with basis
 over Univariate Laurent Polynomial Ring in q over Rational Field and
 Category of filtered algebras
 over Univariate Laurent Polynomial Ring in q over Rational Field and
 Category of commutative algebras
 over Univariate Laurent Polynomial Ring in q over Rational Field and
 Category of realizations of unital magmas]

class Binomial(A)[source]¶

Bases: CombinatorialFreeModule, BindableClass

The quantum-valued polynomial ring in the binomial basis.

The basis used here is given by $B [i] = {[\binom{x}{i}]}_{q}$ for $i \in N$ .

Assuming $n_{1} \leq n_{2}$ , the product of two monomials $B [n_{1}] \cdot B [n_{2}]$ is given by the sum

\sum_{k = 0}^{n_{1}} q^{(k - n_{1}) (k - n_{2})} {[\binom{n_{1}}{k}]}_{q} {[\binom{n_{1} + n_{2} - k}{n_{1}}]}_{q} B [n_{1} + n_{2} - k] .

class Element[source]¶

Bases: IndexedFreeModuleElement

variable_shift(k=1)[source]¶

Return the image by the shift of variables.

On polynomials, the action for $k = 1$ is the shift on variables $x \mapsto 1 + q x$ .

This implementation follows formula (5.5) in [HaHo2017].

INPUT:

$k$ – nonnegative integer (default: 1)

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).B()
sage: B = A.basis()
sage: B[5].variable_shift()
B[4] + q^5*B[5]

product_on_basis(n1, n2)[source]¶

Return the product of basis elements n1 and n2.

INPUT:

n1, n2 – integers

The formula is taken from Theorem 3.4 in [HaHo2017].

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).B()
sage: A.product_on_basis(0, 1)
B[1]

S[source]¶: alias of Shifted

class Shifted(A)[source]¶

Bases: CombinatorialFreeModule, BindableClass

The quantum-valued polynomial ring in the shifted basis.

The basis used here is given by $S [i] = {[\binom{i + x}{i}]}_{q}$ for $i \in N$ .

Assuming $n_{1} \leq n_{2}$ , the product of two monomials $S [n_{1}] \cdot S [n_{2}]$ is given by the sum

\sum_{k = 0}^{n_{1}} (- 1)^{k} q^{(\binom{k}{2}) - n_{1} * n_{2}} {[\binom{n_{1}}{k}]}_{q} {[\binom{n_{1} + n_{2} - k}{n_{1}}]}_{q} S [n_{1} + n_{2} - k] .

class Element[source]¶

Bases: IndexedFreeModuleElement

derivative_at_minus_one()[source]¶

Return the ‘derivative’ at -1.

See also

umbra()

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).derivative_at_minus_one()
1

fraction()[source]¶

Return the generating series of values as a fraction.

See also

h_vector(), h_polynomial()

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: ex = A.basis()[4]
sage: ex.fraction().factor()
(-1) * (t - 1)^-1 * (q*t - 1)^-1 * (q^2*t - 1)^-1 * (q^3*t - 1)^-1 * (q^4*t - 1)^-1

sage: q = polygen(QQ,'q')
sage: x = polygen(q.parent(), 'x')
sage: ex = A.from_polynomial((1+q*x)**3)
sage: ex.fraction().factor()
(t - 1)^-1 * (q*t - 1)^-1 * (q^2*t - 1)^-1 * (q^3*t - 1)^-1 * (q^3*t^2 + 2*q^2*t + 2*q*t + 1)
sage: ex.fraction().numerator()
q^3*t^2 + 2*q^2*t + 2*q*t + 1

h_polynomial()[source]¶

Return the $h$ -vector as a polynomial.

See also

h_vector(), fraction()

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S()
sage: q = polygen(ZZ,'q')
sage: x = polygen(q.parent(),'x')
sage: ex = A.from_polynomial((1+q*x)**3)
sage: ex.h_polynomial()
z^3 + (2*q + 2*q^2)*z^2 + q^3*z

h_vector()[source]¶

Return the numerator of the generating series of values.

If self is an Ehrhart polynomial, this is the h-vector.

See also

h_polynomial(), fraction()

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S()
sage: ex = A.basis()[4]
sage: ex.h_vector()
(0, 0, 0, 0, 1)

sage: q = polygen(QQ,'q')
sage: x = polygen(q.parent(),'x')
sage: ex = A.from_polynomial((1+q*x)**3)
sage: ex.h_vector()
(0, q^3, 2*q + 2*q^2, 1)

umbra()[source]¶

Return the Bernoulli umbra.

This is the derivative at $- 1$ of the shift by one.

This is related to Carlitz’s $q$ -Bernoulli numbers.

See also

derivative_at_minus_one()

EXAMPLES:

sage: F = QuantumValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).umbra()
(q + 2)/(q + 1)

variable_shift(k=1)[source]¶

Return the image by the shift on variables.

The shift is the substitution operator

x \mapsto q x + 1.

INPUT:

$k$ – integer (default: 1)

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S()
sage: S = A.basis()
sage: S[5].variable_shift()
S[0] + q*S[1] + q^2*S[2] + q^3*S[3] + q^4*S[4] + q^5*S[5]

sage: S[5].variable_shift(-1)
-(q^-5)*S[4] + (q^-5)*S[5]

from_h_vector(hv)[source]¶

Convert from some $h$ -vector.

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(ZZ).S()
sage: B = A.basis()
sage: ex = B[2] + B[3]
sage: A.from_h_vector(ex.h_vector())
S[2] + S[3]

sage: q = A.base_ring().gen()
sage: ex = B[2] + q*B[3]
sage: A.from_h_vector(ex.h_vector())
S[2] + q*S[3]

product_on_basis(n1, n2)[source]¶

Return the product of basis elements n1 and n2.

INPUT:

n1, n2 – integers

EXAMPLES:

sage: A = QuantumValuedPolynomialRing(QQ).S()
sage: A.product_on_basis(0, 1)
S[1]

a_realization()[source]¶

Return a default realization.

The Shifted realization is chosen.

EXAMPLES:

sage: QuantumValuedPolynomialRing(QQ).a_realization()
Quantum-Valued Polynomial Ring over Rational Field
in the shifted basis

sage.rings.polynomial.q_integer_valued_polynomials.q_binomial_x(m, n, q=None)[source]¶

Return a $q$ -analogue of binomial(m + x, n).

When evaluated at the $q$ -integer $[k]_{q}$ , this gives the usual $q$ -binomial coefficient $[m + k, n]_{q}$ .

INPUT:

m and n – positive integers
q – optional variable

EXAMPLES:

sage: from sage.combinat.q_analogues import q_int
sage: from sage.rings.polynomial.q_integer_valued_polynomials import q_binomial_x, q_int_x
sage: q_binomial_x(4,2)(0) == q_binomial(4,2)
True
sage: q_binomial_x(3,2)(1) == q_binomial(4,2)
True
sage: q_binomial_x(3,1) == q_int_x(4)
True
sage: q_binomial_x(2,0).parent()
Univariate Polynomial Ring in x over Fraction Field of
Univariate Polynomial Ring in q over Integer Ring

sage.rings.polynomial.q_integer_valued_polynomials.q_int_x(n, q=None)[source]¶

Return the interpolating polynomial of $q$ -integers.

INPUT:

n – a positive integer
q – optional variable

EXAMPLES:

sage: from sage.rings.polynomial.q_integer_valued_polynomials import q_int_x
sage: q_int_x(3)
q^2*x + q + 1