Elements of Infinite Polynomial Rings

AUTHORS:

• Simon King <simon.king@nuigalway.ie>

• Mike Hansen <mhansen@gmail.com>

An Infinite Polynomial Ring has generators \(x_\ast, y_\ast,...\), so that the variables are of the form \(x_0, x_1, x_2, ..., y_0, y_1, y_2,...,...\) (see infinite_polynomial_ring). Using the generators, we can create elements as follows:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: a = x[3]
sage: b = y[4]
sage: a
x_3
sage: b
y_4
sage: c = a*b + a^3 - 2*b^4
sage: c
x_3^3 + x_3*y_4 - 2*y_4^4

Any Infinite Polynomial Ring X is equipped with a monomial ordering. We only consider monomial orderings in which:

X.gen(i)[m] > X.gen(j)[n] \(\iff\) i<j, or i==j and m>n

Under this restriction, the monomial ordering can be lexicographic (default), degree lexicographic, or degree reverse lexicographic. Here, the ordering is lexicographic, and elements can be compared as usual:

sage: X._order
'lex'
sage: a > b
True

Note that, when a method is called that is not directly implemented for ‘InfinitePolynomial’, it is tried to call this method for the underlying classical polynomial. This holds, e.g., when applying the latex function:

sage: latex(c)
x_{3}^{3} + x_{3} y_{4} - 2 y_{4}^{4}

There is a permutation action on Infinite Polynomial Rings by permuting the indices of the variables:

sage: P = Permutation(((4,5),(2,3)))
sage: c^P
x_2^3 + x_2*y_5 - 2*y_5^4

Note that P(0)==0, and thus variables of index zero are invariant under the permutation action. More generally, if P is any callable object that accepts non-negative integers as input and returns non-negative integers, then c^P means to apply P to the variable indices occurring in c.

If you want to substitute variables you can use the standard polynomial methods, such as subs():

sage: R.<x,y> = InfinitePolynomialRing(QQ)
sage: f = x[1] + x[1]*x[2]*x[3]
sage: f.subs({x[1]: x[0]})
x_3*x_2*x_0 + x_0
sage: g = x[0] + x[1] + y[0]
sage: g.subs({x[0]: y[0]})
x_1 + 2*y_0

class sage.rings.polynomial.infinite_polynomial_element.InfinitePolynomial(A, p)

Bases: CommutativePolynomial

Create an element of a Polynomial Ring with a Countably Infinite Number of Variables.

Usually, an InfinitePolynomial is obtained by using the generators of an Infinite Polynomial Ring (see infinite_polynomial_ring) or by conversion.

INPUT:

• A – an Infinite Polynomial Ring.

• p – a classical polynomial that can be interpreted in A.

ASSUMPTIONS:

In the dense implementation, it must be ensured that the argument p coerces into A._P by a name preserving conversion map.

In the sparse implementation, in the direct construction of an infinite polynomial, it is not tested whether the argument p makes sense in A.

EXAMPLES:

sage: from sage.rings.polynomial.infinite_polynomial_element import InfinitePolynomial
sage: X.<alpha> = InfinitePolynomialRing(ZZ)
sage: P.<alpha_1,alpha_2> = ZZ[]

Currently, P and X._P (the underlying polynomial ring of X) both have two variables:

sage: X._P
Multivariate Polynomial Ring in alpha_1, alpha_0 over Integer Ring

By default, a coercion from P to X._P would not be name preserving. However, this is taken care for; a name preserving conversion is impossible, and by consequence an error is raised:

sage: InfinitePolynomial(X, (alpha_1+alpha_2)^2)
Traceback (most recent call last):
...
TypeError: Could not find a mapping of the passed element to this ring.

When extending the underlying polynomial ring, the construction of an infinite polynomial works:

sage: alpha[2]
alpha_2
sage: InfinitePolynomial(X, (alpha_1+alpha_2)^2)
alpha_2^2 + 2*alpha_2*alpha_1 + alpha_1^2

In the sparse implementation, it is not checked whether the polynomial really belongs to the parent, and when it does not, the results may be unexpected due to coercions:

sage: Y.<alpha,beta> = InfinitePolynomialRing(GF(2), implementation='sparse')
sage: a = (alpha_1+alpha_2)^2
sage: InfinitePolynomial(Y, a)
alpha_0^2 + beta_0^2

However, it is checked when doing a conversion:

sage: Y(a)
alpha_2^2 + alpha_1^2

coefficient(monomial)

Returns the coefficient of a monomial in this polynomial.

INPUT:

• A monomial (element of the parent of self) or

• a dictionary that describes a monomial (the keys are variables of the parent of self, the values are the corresponding exponents)

EXAMPLES:

We can get the coefficient in front of monomials:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: a = 2*x[0]*x[1] + x[1] + x[2]
sage: a.coefficient(x[0])
2*x_1
sage: a.coefficient(x[1])
2*x_0 + 1
sage: a.coefficient(x[2])
1
sage: a.coefficient(x[0]*x[1])
2

We can also pass in a dictionary:

sage: a.coefficient({x[0]:1, x[1]:1})
2

footprint()

Leading exponents sorted by index and generator.

OUTPUT:

D – a dictionary whose keys are the occurring variable indices.

D[s] is a list [i_1,...,i_n], where i_j gives the exponent of self.parent().gen(j)[s] in the leading term of self.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = x[30]*y[1]^3*x[1]^2 + 2*x[10]*y[30]
sage: sorted(p.footprint().items())
[(1, [2, 3]), (30, [1, 0])]

gcd(x)

computes the greatest common divisor

EXAMPLES:

sage: R.<x>=InfinitePolynomialRing(QQ)
sage: p1=x[0] + x[1]**2
sage: gcd(p1,p1+3)
1
sage: gcd(p1,p1)==p1
True

is_nilpotent()

Return True if self is nilpotent, i.e., some power of self is 0.

EXAMPLES:

sage: R.<x> = InfinitePolynomialRing(QQbar)                                 # needs sage.rings.number_field
sage: (x[0] + x[1]).is_nilpotent()                                          # needs sage.rings.number_field
False
sage: R(0).is_nilpotent()                                                   # needs sage.rings.number_field
True
sage: _.<x> = InfinitePolynomialRing(Zmod(4))
sage: (2*x[0]).is_nilpotent()
True
sage: (2+x[4]*x[7]).is_nilpotent()
False
sage: _.<y> = InfinitePolynomialRing(Zmod(100))
sage: (5+2*y[0] + 10*(y[0]^2+y[1]^2)).is_nilpotent()
False
sage: (10*y[2] + 20*y[5] - 30*y[2]*y[5] + 70*(y[2]^2+y[5]^2)).is_nilpotent()
True

is_unit()

Answer whether self is a unit.

EXAMPLES:

sage: R1.<x,y> = InfinitePolynomialRing(ZZ)
sage: R2.<a,b> = InfinitePolynomialRing(QQ)
sage: (1 + x[2]).is_unit()
False
sage: R1(1).is_unit()
True
sage: R1(2).is_unit()
False
sage: R2(2).is_unit()
True
sage: (1 + a[2]).is_unit()
False

Check that github issue #22454 is fixed:

sage: _.<x> = InfinitePolynomialRing(Zmod(4))
sage: (1 + 2*x[0]).is_unit()
True
sage: (x[0]*x[1]).is_unit()
False
sage: _.<x> = InfinitePolynomialRing(Zmod(900))
sage: (7+150*x[0] + 30*x[1] + 120*x[1]*x[100]).is_unit()
True

lc()

The coefficient of the leading term of self.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = 2*x[10]*y[30] + 3*x[10]*y[1]^3*x[1]^2
sage: p.lc()
3

lm()

The leading monomial of self.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = 2*x[10]*y[30] + x[10]*y[1]^3*x[1]^2
sage: p.lm()
x_10*x_1^2*y_1^3

lt()

The leading term (= product of coefficient and monomial) of self.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = 2*x[10]*y[30] + 3*x[10]*y[1]^3*x[1]^2
sage: p.lt()
3*x_10*x_1^2*y_1^3

max_index()

Return the maximal index of a variable occurring in self, or -1 if self is scalar.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = x[1]^2 + y[2]^2 + x[1]*x[2]*y[3] + x[1]*y[4]
sage: p.max_index()
4
sage: x[0].max_index()
0
sage: X(10).max_index()
-1

polynomial()

Return the underlying polynomial.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(GF(7))
sage: p = x[2]*y[1] + 3*y[0]
sage: p
x_2*y_1 + 3*y_0
sage: p.polynomial()
x_2*y_1 + 3*y_0
sage: p.polynomial().parent()
Multivariate Polynomial Ring in x_2, x_1, x_0, y_2, y_1, y_0
 over Finite Field of size 7
sage: p.parent()
Infinite polynomial ring in x, y over Finite Field of size 7

reduce(I, tailreduce=False, report=None)

Symmetrical reduction of self with respect to a symmetric ideal (or list of Infinite Polynomials).

INPUT:

• I – a SymmetricIdeal or a list of Infinite Polynomials.

• tailreduce – (bool, default False) Tail reduction is performed if this parameter is True.

• report – (object, default None) If not None, some information on the progress of computation is printed, since reduction of huge polynomials may take a long time.

OUTPUT:

Symmetrical reduction of self with respect to I, possibly with tail reduction.

THEORY:

Reducing an element \(p\) of an Infinite Polynomial Ring \(X\) by some other element \(q\) means the following:

1. Let \(M\) and \(N\) be the leading terms of \(p\) and \(q\).

2. Test whether there is a permutation \(P\) that does not does not diminish the variable indices occurring in \(N\) and preserves their order, so that there is some term \(T\in X\) with \(TN^P = M\). If there is no such permutation, return \(p\)

3. Replace \(p\) by \(p-T q^P\) and continue with step 1.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = y[1]^2*y[3] + y[2]*x[3]^3
sage: p.reduce([y[2]*x[1]^2])
x_3^3*y_2 + y_3*y_1^2

The preceding is correct: If a permutation turns y[2]*x[1]^2 into a factor of the leading monomial y[2]*x[3]^3 of p, then it interchanges the variable indices 1 and 2; this is not allowed in a symmetric reduction. However, reduction by y[1]*x[2]^2 works, since one can change variable index 1 into 2 and 2 into 3:

sage: p.reduce([y[1]*x[2]^2])                                               # needs sage.libs.singular
y_3*y_1^2

The next example shows that tail reduction is not done, unless it is explicitly advised. The input can also be a Symmetric Ideal:

sage: I = (y[3])*X
sage: p.reduce(I)
x_3^3*y_2 + y_3*y_1^2
sage: p.reduce(I, tailreduce=True)                                          # needs sage.libs.singular
x_3^3*y_2

Last, we demonstrate the report option:

sage: p = x[1]^2 + y[2]^2 + x[1]*x[2]*y[3] + x[1]*y[4]
sage: p.reduce(I, tailreduce=True, report=True)                             # needs sage.libs.singular
:T[2]:>
>
x_1^2 + y_2^2

The output ‘:’ means that there was one reduction of the leading monomial. ‘T[2]’ means that a tail reduction was performed on a polynomial with two terms. At ‘>’, one round of the reduction process is finished (there could only be several non-trivial rounds if \(I\) was generated by more than one polynomial).

ring()

The ring which self belongs to.

This is the same as self.parent().

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(ZZ,implementation='sparse')
sage: p = x[100]*y[1]^3*x[1]^2 + 2*x[10]*y[30]
sage: p.ring()
Infinite polynomial ring in x, y over Integer Ring

squeezed()

Reduce the variable indices occurring in self.

OUTPUT:

Apply a permutation to self that does not change the order of the variable indices of self but squeezes them into the range 1,2,…

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ,implementation='sparse')
sage: p = x[1]*y[100] + x[50]*y[1000]
sage: p.squeezed()
x_2*y_4 + x_1*y_3

stretch(k)

Stretch self by a given factor.

INPUT:

k – an integer.

OUTPUT:

Replace \(v_n\) with \(v_{n\cdot k}\) for all generators \(v_\ast\) occurring in self.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: a = x[0] + x[1] + x[2]
sage: a.stretch(2)
x_4 + x_2 + x_0

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: a = x[0] + x[1] + y[0]*y[1]; a
x_1 + x_0 + y_1*y_0
sage: a.stretch(2)
x_2 + x_0 + y_2*y_0

subs(fixed=None, **kwargs)

Substitute variables in self.

INPUT:

• fixed – (optional) dict with {variable: value} pairs

• **kwargs – named parameters

OUTPUT:

the resulting substitution

EXAMPLES:

sage: R.<x,y> = InfinitePolynomialRing(QQ)
sage: f = x[1] + x[1]*x[2]*x[3]

Passing fixed={x[1]: x[0]}. Note that the keys may be given using the generators of the infinite polynomial ring or as a string:

sage: f.subs({x[1]: x[0]})
x_3*x_2*x_0 + x_0
sage: f.subs({'x_1': x[0]})
x_3*x_2*x_0 + x_0

Passing the variables as names parameters:

sage: f.subs(x_1=y[1])
x_3*x_2*y_1 + y_1
sage: f.subs(x_1=y[1], x_2=2)
2*x_3*y_1 + y_1

The substitution returns the original polynomial if you try to substitute a variable not present:

sage: g = x[0] + x[1]
sage: g.subs({y[0]: x[0]})
x_1 + x_0

The substitution can also handle matrices:

sage: # needs sage.modules
sage: M = matrix([[1,0], [0,2]])
sage: N = matrix([[0,3], [4,0]])
sage: g = x[0]^2 + 3*x[1]
sage: g.subs({'x_0': M})
[3*x_1 + 1         0]
[        0 3*x_1 + 4]
sage: g.subs({x[0]: M, x[1]: N})
[ 1  9]
[12  4]

If you pass both fixed and kwargs, any conflicts will defer to fixed:

sage: R.<x,y> = InfinitePolynomialRing(QQ)
sage: f = x[0]
sage: f.subs({x[0]: 1})
1
sage: f.subs(x_0=5)
5
sage: f.subs({x[0]: 1}, x_0=5)
1

symmetric_cancellation_order(other)

Comparison of leading terms by Symmetric Cancellation Order, \(<_{sc}\).

INPUT:

self, other – two Infinite Polynomials

ASSUMPTION:

Both Infinite Polynomials are non-zero.

OUTPUT:

(c, sigma, w), where

• c = -1,0,1, or None if the leading monomial of self is smaller, equal, greater, or incomparable with respect to other in the monomial ordering of the Infinite Polynomial Ring

• sigma is a permutation witnessing self \(<_{sc}\) other (resp. self \(>_{sc}\) other) or is 1 if self.lm()==other.lm()

• w is 1 or is a term so that w*self.lt()^sigma == other.lt() if \(c\le 0\), and w*other.lt()^sigma == self.lt() if \(c=1\)

THEORY:

If the Symmetric Cancellation Order is a well-quasi-ordering then computation of Groebner bases always terminates. This is the case, e.g., if the monomial order is lexicographic. For that reason, lexicographic order is our default order.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: (x[2]*x[1]).symmetric_cancellation_order(x[2]^2)
(None, 1, 1)
sage: (x[2]*x[1]).symmetric_cancellation_order(x[2]*x[3]*y[1])
(-1, [2, 3, 1], y_1)
sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[1])
(None, 1, 1)
sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[2])
(-1, [2, 3, 1], 1)

tail()

The tail of self (this is self minus its leading term).

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: p = 2*x[10]*y[30] + 3*x[10]*y[1]^3*x[1]^2
sage: p.tail()
2*x_10*y_30

variables()

Return the variables occurring in self (tuple of elements of some polynomial ring).

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: p = x[1] + x[2] - 2*x[1]*x[3]
sage: p.variables()
(x_3, x_2, x_1)
sage: x[1].variables()
(x_1,)
sage: X(1).variables()
()

class sage.rings.polynomial.infinite_polynomial_element.InfinitePolynomial_dense(A, p)

Bases: InfinitePolynomial

Element of a dense Polynomial Ring with a Countably Infinite Number of Variables.

INPUT:

• A – an Infinite Polynomial Ring in dense implementation

• p – a classical polynomial that can be interpreted in A.

Of course, one should not directly invoke this class, but rather construct elements of A in the usual way.

class sage.rings.polynomial.infinite_polynomial_element.InfinitePolynomial_sparse(A, p)

Bases: InfinitePolynomial

Element of a sparse Polynomial Ring with a Countably Infinite Number of Variables.

INPUT:

• A – an Infinite Polynomial Ring in sparse implementation

• p – a classical polynomial that can be interpreted in A.

Of course, one should not directly invoke this class, but rather construct elements of A in the usual way.

EXAMPLES:

sage: A.<a> = QQ[]
sage: B.<b,c> = InfinitePolynomialRing(A,implementation='sparse')
sage: p = a*b[100] + 1/2*c[4]
sage: p
a*b_100 + 1/2*c_4
sage: p.parent()
Infinite polynomial ring in b, c
 over Univariate Polynomial Ring in a over Rational Field
sage: p.polynomial().parent()
Multivariate Polynomial Ring in b_100, b_0, c_4, c_0
 over Univariate Polynomial Ring in a over Rational Field