\(p\)-Adic Generic Nodes

This file contains a bunch of intermediate classes for the \(p\)-adic parents, allowing a function to be implemented at the right level of generality.

AUTHORS:

• David Roe

class sage.rings.padics.generic_nodes.CappedAbsoluteGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

is_capped_absolute()

Return whether this \(p\)-adic ring bounds precision in a capped absolute fashion.

The absolute precision of an element is the power of \(p\) modulo which that element is defined. In a capped absolute ring, the absolute precision of elements are bounded by a constant depending on the ring.

EXAMPLES:

sage: R = ZpCA(5, 15)
sage: R.is_capped_absolute()
True
sage: R(5^7)
5^7 + O(5^15)
sage: S = Zp(5, 15)
sage: S.is_capped_absolute()
False
sage: S(5^7)
5^7 + O(5^22)

class sage.rings.padics.generic_nodes.CappedRelativeFieldGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

class sage.rings.padics.generic_nodes.CappedRelativeGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

is_capped_relative()

Return whether this \(p\)-adic ring bounds precision in a capped relative fashion.

The relative precision of an element is the power of p modulo which the unit part of that element is defined. In a capped relative ring, the relative precision of elements are bounded by a constant depending on the ring.

EXAMPLES:

sage: R = ZpCA(5, 15)
sage: R.is_capped_relative()
False
sage: R(5^7)
5^7 + O(5^15)
sage: S = Zp(5, 15)
sage: S.is_capped_relative()
True
sage: S(5^7)
5^7 + O(5^22)

class sage.rings.padics.generic_nodes.CappedRelativeRingGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.CappedRelativeGeneric

class sage.rings.padics.generic_nodes.FixedModGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

is_fixed_mod()

Return whether this \(p\)-adic ring bounds precision in a fixed modulus fashion.

The absolute precision of an element is the power of p modulo which that element is defined. In a fixed modulus ring, the absolute precision of every element is defined to be the precision cap of the parent. This means that some operations, such as division by \(p\), don’t return a well defined answer.

EXAMPLES:

sage: R = ZpFM(5,15)
sage: R.is_fixed_mod()
True
sage: R(5^7,absprec=9)
5^7
sage: S = ZpCA(5, 15)
sage: S.is_fixed_mod()
False
sage: S(5^7,absprec=9)
5^7 + O(5^9)

class sage.rings.padics.generic_nodes.FloatingPointFieldGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.FloatingPointGeneric

class sage.rings.padics.generic_nodes.FloatingPointGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.local_generic.LocalGeneric

is_floating_point()

Return whether this \(p\)-adic ring uses a floating point precision model.

Elements in the floating point model are stored by giving a valuation and a unit part. Arithmetic is done where the unit part is truncated modulo a fixed power of the uniformizer, stored in the precision cap of the parent.

EXAMPLES:

sage: R = ZpFP(5,15)
sage: R.is_floating_point()
True
sage: R(5^7,absprec=9)
5^7
sage: S = ZpCR(5,15)
sage: S.is_floating_point()
False
sage: S(5^7,absprec=9)
5^7 + O(5^9)

class sage.rings.padics.generic_nodes.FloatingPointRingGeneric(base, prec, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.FloatingPointGeneric

sage.rings.padics.generic_nodes.is_pAdicField(R)

Return True if and only if R is a \(p\)-adic field.

EXAMPLES:

sage: is_pAdicField(Zp(17))
False
sage: is_pAdicField(Qp(17))
True

sage.rings.padics.generic_nodes.is_pAdicRing(R)

Return True if and only if R is a \(p\)-adic ring (not a field).

EXAMPLES:

sage: is_pAdicRing(Zp(5))
True
sage: is_pAdicRing(RR)
False

class sage.rings.padics.generic_nodes.pAdicCappedAbsoluteRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.CappedAbsoluteGeneric

class sage.rings.padics.generic_nodes.pAdicCappedRelativeFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicFieldGeneric, sage.rings.padics.generic_nodes.CappedRelativeFieldGeneric

class sage.rings.padics.generic_nodes.pAdicCappedRelativeRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.CappedRelativeRingGeneric

class sage.rings.padics.generic_nodes.pAdicFieldBaseGeneric(p, prec, print_mode, names, element_class)

Bases: sage.rings.padics.padic_base_generic.pAdicBaseGeneric, sage.rings.padics.generic_nodes.pAdicFieldGeneric

composite(subfield1, subfield2)

Return the composite of two subfields of self, i.e., the largest subfield containing both

INPUT:

• self – a \(p\)-adic field

• subfield1 – a subfield

• subfield2 – a subfield

OUTPUT:

• the composite of subfield1 and subfield2

EXAMPLES:

sage: K = Qp(17); K.composite(K, K) is K
True

construction(forbid_frac_field=False)

Return the functorial construction of self, namely, completion of the rational numbers with respect a given prime.

Also preserves other information that makes this field unique (e.g. precision, rounding, print mode).

INPUT:

• forbid_frac_field – require a completion functor rather than a fraction field functor. This is used in the sage.rings.padics.local_generic.LocalGeneric.change() method.

EXAMPLES:

sage: K = Qp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
17-adic Ring with capped relative precision 8
sage: c
FractionField
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True

We can get a completion functor by forbidding the fraction field:

sage: c, L = K.construction(forbid_frac_field=True); L
Rational Field
sage: c
Completion[17, prec=8]
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True

subfield(list)

Return the subfield generated by the elements in list

INPUT:

• self – a \(p\)-adic field

• list – a list of elements of self

OUTPUT:

• the subfield of self generated by the elements of list

EXAMPLES:

sage: K = Qp(17); K.subfield([K(17), K(1827)]) is K
True

subfields_of_degree(n)

Return the number of subfields of self of degree \(n\)

INPUT:

• self – a \(p\)-adic field

• n – an integer

OUTPUT:

• integer – the number of subfields of degree n over self.base_ring()

EXAMPLES:

sage: K = Qp(17)
sage: K.subfields_of_degree(1)
1

class sage.rings.padics.generic_nodes.pAdicFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric, sage.rings.ring.Field

class sage.rings.padics.generic_nodes.pAdicFixedModRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.FixedModGeneric

class sage.rings.padics.generic_nodes.pAdicFloatingPointFieldGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicFieldGeneric, sage.rings.padics.generic_nodes.FloatingPointFieldGeneric

class sage.rings.padics.generic_nodes.pAdicFloatingPointRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.generic_nodes.pAdicRingGeneric, sage.rings.padics.generic_nodes.FloatingPointRingGeneric

class sage.rings.padics.generic_nodes.pAdicLatticeGeneric(p, prec, print_mode, names, label=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric

An implementation of the \(p\)-adic rationals with lattice precision.

INPUT:

• \(p\) – the underlying prime number

• prec – the precision

• subtype – either "cap" or "float", specifying the precision model used for tracking precision

• label – a string or None (default: None)

convert_multiple(*elts)

Convert a list of elements to this parent.

NOTE:

This function tries to be sharp on precision as much as possible. In particular, if the precision of the input elements are handled by a lattice, diffused digits of precision are preserved during the conversion.

EXAMPLES:

sage: R = ZpLC(2)
sage: x = R(1, 10); y = R(1, 5)
sage: x,y = x+y, x-y

Remark that the pair \((x,y)\) has diffused digits of precision:

sage: x
2 + O(2^5)
sage: y
O(2^5)
sage: x + y
2 + O(2^11)

sage: R.precision().diffused_digits([x,y])
6

As a consequence, if we convert x and y separately, we loose some precision:

sage: R2 = ZpLC(2, label='copy')
sage: x2 = R2(x); y2 = R2(y)
sage: x2
2 + O(2^5)
sage: y2
O(2^5)
sage: x2 + y2
2 + O(2^5)

sage: R2.precision().diffused_digits([x2,y2])
0

On the other hand, this issue disappears when we use multiple conversion:

sage: x2,y2 = R2.convert_multiple(x,y)
sage: x2 + y2
2 + O(2^11)

sage: R2.precision().diffused_digits([x2,y2])
6

is_lattice_prec()

Return whether this \(p\)-adic ring bounds precision using a lattice model.

In lattice precision, relationships between elements are stored in a precision object of the parent, which allows for optimal precision tracking at the cost of increased memory usage and runtime.

EXAMPLES:

sage: R = ZpCR(5, 15)
sage: R.is_lattice_prec()
False
sage: x = R(25, 8)
sage: x - x
O(5^8)
sage: S = ZpLC(5, 15)
sage: S.is_lattice_prec()
True
sage: x = S(25, 8)
sage: x - x
O(5^30)

label()

Return the label of this parent.

NOTE:

Labels can be used to distinguish between parents with the same defining data.

They are useful in the lattice precision framework in order to limit the size of the lattice modeling the precision (which is roughly the number of elements having this parent).

Elements of a parent with some label do not coerce to a parent with a different label. However conversions are allowed.

EXAMPLES:

sage: R = ZpLC(5)
sage: R.label()  # no label by default

sage: R = ZpLC(5, label='mylabel')
sage: R.label()
'mylabel'

Labels are typically useful to isolate computations. For example, assume that we first want to do some calculations with matrices:

sage: R = ZpLC(5, label='matrices')
sage: M = random_matrix(R, 4, 4)
sage: d = M.determinant()

Now, if we want to do another unrelated computation, we can use a different label:

sage: R = ZpLC(5, label='polynomials')
sage: S.<x> = PolynomialRing(R)
sage: P = (x-1)*(x-2)*(x-3)*(x-4)*(x-5)

Without labels, the software would have modeled the precision on the matrices and on the polynomials using the same lattice (manipulating a lattice of higher dimension can have a significant impact on performance).

precision()

Return the lattice precision object attached to this parent.

EXAMPLES:

sage: R = ZpLC(5, label='precision')
sage: R.precision()
Precision lattice on 0 objects (label: precision)

sage: x = R(1, 10); y = R(1, 5)
sage: R.precision()
Precision lattice on 2 objects (label: precision)

See also

sage.rings.padics.lattice_precision.PrecisionLattice

precision_cap()

Return the relative precision cap for this ring if it is finite. Otherwise return the absolute precision cap.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap()
20
sage: R.precision_cap_relative()
20

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap()
20
sage: R.precision_cap_relative()
+Infinity
sage: R.precision_cap_absolute()
20

See also

precision_cap_relative(), precision_cap_absolute()

precision_cap_absolute()

Return the absolute precision cap for this ring.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap_absolute()
40

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap_absolute()
20

See also

precision_cap(), precision_cap_relative()

precision_cap_relative()

Return the relative precision cap for this ring.

EXAMPLES:

sage: R = ZpLC(3)
sage: R.precision_cap_relative()
20

sage: R = ZpLC(3, prec=(infinity,20))
sage: R.precision_cap_relative()
+Infinity

See also

precision_cap(), precision_cap_absolute()

class sage.rings.padics.generic_nodes.pAdicRelaxedGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric

Generic class for relaxed \(p\)-adics.

INPUT:

• \(p\) – the underlying prime number

• prec – the default precision

an_element(unbounded=False)

Return an element in this ring.

EXAMPLES:

sage: R = ZpER(7, prec=5)
sage: R.an_element()
7 + O(7^5)
sage: R.an_element(unbounded=True)
7 + ...

default_prec()

Return the default precision of this relaxed \(p\)-adic ring.

The default precision is mostly used for printing: it is the number of digits which are printed for unbounded elements (that is elements having infinite absolute precision).

EXAMPLES:

sage: R = ZpER(5, print_mode="digits")
sage: R.default_prec()
20
sage: R(1/17)
...34024323104201213403

sage: S = ZpER(5, prec=10, print_mode="digits")
sage: S.default_prec()
10
sage: S(1/17)
...4201213403

halting_prec()

Return the default halting precision of this relaxed \(p\)-adic ring.

The halting precision is the precision at which elements of this parent are compared (unless more digits have been previously computed). By default, it is twice the default precision.

EXAMPLES:

sage: R = ZpER(5, print_mode="digits")
sage: R.halting_prec()
40

is_relaxed()

Return whether this \(p\)-adic ring is relaxed.

EXAMPLES:

sage: R = Zp(5)
sage: R.is_relaxed()
False
sage: S = ZpER(5)
sage: S.is_relaxed()
True

is_secure()

Return False if this \(p\)-adic relaxed ring is not secure (i.e. if indistinguishable elements at the working precision are considered as equal); True otherwise (in which case, an error is raised when equality cannot be decided).

EXAMPLES:

sage: R = ZpER(5)
sage: R.is_secure()
False
sage: x = R(20/21)
sage: y = x + 5^50
sage: x == y
True

sage: S = ZpER(5, secure=True)
sage: S.is_secure()
True
sage: x = S(20/21)
sage: y = x + 5^50
sage: x == y
Traceback (most recent call last):
...
PrecisionError: unable to decide equality; try to bound precision

precision_cap()

Return the precision cap of this \(p\)-adic ring, which is infinite in the case of relaxed rings.

EXAMPLES:

sage: R = ZpER(5)
sage: R.precision_cap()
+Infinity

random_element(integral=False, prec=None)

Return a random element in this ring.

INPUT:

• integral – a boolean (default: False); if True, return a random element in the ring of integers of this ring

• prec – an integer or None (default: None); if given, bound the precision of the output to prec

EXAMPLES:

sage: R = ZpER(5, prec=10)

By default, this method returns a unbounded element:

sage: a = R.random_element()
sage: a  # random
4 + 3*5 + 3*5^2 + 5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 5^7 + 5^9 + ...
sage: a.precision_absolute()
+Infinity

The precision can be bounded by passing in a precision:

sage: b = R.random_element(prec=15)
sage: b  # random
2 + 3*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6 + 3*5^8 + 3*5^9 + 4*5^10 + 5^11 + 4*5^12 + 5^13 + 2*5^14 + O(5^15)
sage: b.precision_absolute()
15

some_elements(unbounded=False)

Return a list of elements in this ring.

This is typically used for running generic tests (see TestSuite).

EXAMPLES:

sage: R = ZpER(7, prec=5)
sage: R.some_elements()
[O(7^5),
 1 + O(7^5),
 7 + O(7^5),
 7 + O(7^5),
 1 + 5*7 + 3*7^2 + 6*7^3 + O(7^5),
 7 + 6*7^2 + 6*7^3 + 6*7^4 + O(7^5)]

sage: R.some_elements(unbounded=True)
[0,
 1 + ...,
 7 + ...,
 7 + ...,
 1 + 5*7 + 3*7^2 + 6*7^3 + ...,
 7 + 6*7^2 + 6*7^3 + 6*7^4 + ...]

teichmuller(x)

Return the Teichmuller representative of \(x\).

EXAMPLES:

sage: R = ZpER(5, print_mode="digits")
sage: R.teichmuller(2)
...40423140223032431212

teichmuller_system()

Return a set of teichmuller representatives for the invertible elements of \(\ZZ / p\ZZ\).

EXAMPLES:

sage: R = ZpER(7, print_mode="digits")
sage: R.teichmuller_system()
[...00000000000000000001,
 ...16412125443426203642,
 ...16412125443426203643,
 ...50254541223240463024,
 ...50254541223240463025,
 ...66666666666666666666]

unknown(start_val=0, digits=None)

Return a self-referent number in this ring.

INPUT:

• start_val – an integer (default: 0); a lower bound on the valuation of the returned element

• digits – an element, a list or None (default: None); the first digit or the list of the digits of the returned element

NOTE:

Self-referent numbers are numbers whose digits are defined in terms of the previous ones. This method is used to declare a self-referent number (and optionally, to set its first digits). The definition of the number itself will be given afterwords using to method meth:\(sage.rings.padics.relaxed_template.RelaxedElement_unknown.set\) of the element.

EXAMPLES:

sage: R = ZpER(5, prec=10)

We declare a self-referent number:

sage: a = R.unknown()

So far, we do not know anything on \(a\) (except that it has nonnegative valuation):

sage: a
O(5^0)

We can now use the method meth:\(sage.rings.padics.relaxed_template.RelaxedElement_unknown.set\) to define \(a\). Below, for example, we say that the digits of \(a\) have to agree with the digits of \(1 + 5 a\). Note that the factor \(5\) shifts the digits; the \(n\)-th digit of \(a\) is then defined by the previous ones:

sage: a.set(1 + 5*a)
True

After this, \(a\) contains the solution of the equation \(a = 1 + 5 a\), that is \(a = -1/4\):

sage: a
1 + 5 + 5^2 + 5^3 + 5^4 + 5^5 + 5^6 + 5^7 + 5^8 + 5^9 + ...

Here is another example with an equation of degree \(2\):

sage: b = R.unknown()
sage: b.set(1 - 5*b^2)
True
sage: b
1 + 4*5 + 5^2 + 3*5^4 + 4*5^6 + 4*5^8 + 2*5^9 + ...
sage: (sqrt(R(21)) - 1) / 10
1 + 4*5 + 5^2 + 3*5^4 + 4*5^6 + 4*5^8 + 2*5^9 + ...

Cross self-referent definitions are also allowed:

sage: u = R.unknown()
sage: v = R.unknown()
sage: w = R.unknown()

sage: u.set(1 + 2*v + 3*w^2 + 5*u*v*w)
True
sage: v.set(2 + 4*w + sqrt(1 + 5*u + 10*v + 15*w))
True
sage: w.set(3 + 25*(u*v + v*w + u*w))
True

sage: u
3 + 3*5 + 4*5^2 + 5^3 + 3*5^4 + 5^5 + 5^6 + 3*5^7 + 5^8 + 3*5^9 + ...
sage: v
4*5 + 2*5^2 + 4*5^3 + 5^4 + 5^5 + 3*5^6 + 5^8 + 5^9 + ...
sage: w
3 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 2*5^6 + 5^8 + 5^9 + ...

class sage.rings.padics.generic_nodes.pAdicRingBaseGeneric(p, prec, print_mode, names, element_class)

Bases: sage.rings.padics.padic_base_generic.pAdicBaseGeneric, sage.rings.padics.generic_nodes.pAdicRingGeneric

construction(forbid_frac_field=False)

Return the functorial construction of self, namely, completion of the rational numbers with respect a given prime.

Also preserves other information that makes this field unique (e.g. precision, rounding, print mode).

INPUT:

• forbid_frac_field – ignored, for compatibility with other p-adic types.

EXAMPLES:

sage: K = Zp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Integer Ring
sage: c(L)
17-adic Ring with capped relative precision 8
sage: K == c(L)
True

random_element(algorithm='default')

Return a random element of self, optionally using the algorithm argument to decide how it generates the element. Algorithms currently implemented:

• default: Choose \(a_i\), \(i >= 0\), randomly between \(0\) and \(p-1\) until a nonzero choice is made. Then continue choosing \(a_i\) randomly between \(0\) and \(p-1\) until we reach precision_cap, and return \(\sum a_i p^i\).

EXAMPLES:

sage: Zp(5,6).random_element()
3 + 3*5 + 2*5^2 + 3*5^3 + 2*5^4 + 5^5 + O(5^6)
sage: ZpCA(5,6).random_element()
4*5^2 + 5^3 + O(5^6)
sage: ZpFM(5,6).random_element()
2 + 4*5^2 + 2*5^4 + 5^5

class sage.rings.padics.generic_nodes.pAdicRingGeneric(base, p, prec, print_mode, names, element_class, category=None)

Bases: sage.rings.padics.padic_generic.pAdicGeneric, sage.rings.ring.EuclideanDomain

is_field(proof=True)

Return whether this ring is actually a field, ie False.

EXAMPLES:

sage: Zp(5).is_field()
False

krull_dimension()

Return the Krull dimension of self, i.e. 1

INPUT:

• self – a \(p\)-adic ring

OUTPUT:

• the Krull dimension of self. Since self is a \(p\)-adic ring, this is 1.

EXAMPLES:

sage: Zp(5).krull_dimension()
1