Struct PadicRing 

pub struct PadicRing { /* private fields */ }

The p-adic integers known modulo p^N, with N fixed at construction.

This is a finite-precision context, not the infinite ring Z_p disguised as exact. Every element is a canonical FLINT-backed residue modulo p^N, so its coefficient storage is bounded before allocation by the context’s Self::modulus. Arithmetic preserves the same absolute precision cap.

Implementations

impl PadicRing

pub fn coefficient_frobenius(&self, exponent: u64) -> PadicRingFrobenius

Checked coefficient Frobenius power, the identity on Z_p coefficients.

impl PadicRing

pub fn precision_restriction( &self, target_absolute_precision: u64, ) -> Result<PadicRingRestriction, PadicRestrictionError>

Construct a first-class restriction to a coarser fixed-cap ring.

pub fn divide_uniformizer_exact( &self, value: &PadicElement, exponent: u64, ) -> Result<PadicElement, PadicUniformizerDivisionError>

Divide by p^exponent, requiring the quotient to remain integral.

impl PadicRing

pub fn new<T: Into<Integer>>( prime: T, absolute_precision: u64, ) -> Result<Self, PadicContextError>

Construct fixed-cap p-adic integers known modulo p^N.

pub fn prime(&self) -> &Integer

Residue characteristic p.

pub fn absolute_precision(&self) -> u64

Immutable absolute precision cap N.

pub fn modulus(&self) -> &Integer

Storage modulus p^N; every element has a representative in [0, p^N), giving the promised per-element coefficient bound.

pub fn residue_field(&self) -> &FiniteField

Prime residue field F_p.

pub fn element<T: Into<Integer>>(&self, value: T) -> PadicElement

Embed an integer as a value known modulo p^N.

pub fn to_json(&self) -> Result<String, NoetherError>

Serialize the fixed prime and absolute precision as canonical typed JSON.

pub fn from_json(value: &str) -> Result<Self, NoetherError>

Deserialize and revalidate a fixed-precision p-adic ring context.

pub fn residue( &self, value: &PadicElement, ) -> Result<FiniteFieldElement, ContextMismatchError>

Reduce a p-adic integer to its prime residue field.

pub fn teichmuller( &self, value: &FiniteFieldElement, ) -> Result<PadicElement, ContextMismatchError>

Lift a residue-field element to its Teichmüller representative modulo p^N, characterized by [a]^p = [a] and [a] mod p = a.

pub fn hensel_lift( &self, polynomial: &Polynomial<Self>, residue_root: &FiniteFieldElement, ) -> Result<PadicElement, HenselLiftError>

Lift a simple root modulo p to a root at this context’s full precision.

The polynomial must be univariate over this exact PadicRing, the supplied value must be a root in its residue field, and the derivative must be nonzero there. The returned value is checked to annihilate the polynomial modulo p^N; unmet hypotheses fail closed.

pub fn restrict( &self, value: &PadicElement, requested_absolute_precision: u64, ) -> Result<PadicElement, PadicRestrictionError>

Discard high p-adic digits explicitly, producing a value in the distinct context (p, requested_absolute_precision).

Trait Implementations

impl Clone for PadicRing

fn clone(&self) -> PadicRing

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl CoefficientFrobeniusContext for PadicRing

type Frobenius = PadicRingFrobenius

Concrete checked Frobenius-map value returned for this context.

fn coefficient_frobenius(&self, exponent: u64) -> Self::Frobenius

Construct the declared nonnegative Frobenius power.

fn residue_characteristic(&self) -> &Integer

Residue characteristic whose power scales monomial exponents.

impl Debug for PadicRing

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl<'de> Deserialize<'de> for PadicRing

fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>

where D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Display for PadicRing

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl FiniteContext for PadicRing

fn cardinality(&self) -> Integer

Exact cardinality without eager enumeration.

fn elements(&self) -> Box<dyn Iterator<Item = Self::Element> + '_>

Deterministic lazy iteration over all elements.

impl PartialEq for PadicRing

fn eq(&self, other: &Self) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl PrecisionContext for PadicRing

type Precision = u64

Domain-specific precision shape.

fn precision(&self) -> &Self::Precision

The immutable precision specification of this context.

fn refines( &self, candidate: &Self::Precision, reference: &Self::Precision, ) -> bool

Whether candidate contains at least the information in reference.

impl Ring for PadicRing

type Element = PadicElement

Elements manufactured and validated by this context.

fn contains(&self, value: &Self::Element) -> bool

Test whether a value belongs to this exact context.

fn zero(&self) -> Self::Element

The additive identity in this context.

fn one(&self) -> Self::Element

The multiplicative identity in this context.

fn add( &self, a: &Self::Element, b: &Self::Element, ) -> Result<Self::Element, ContextMismatchError>

Add two elements after checking their context.

fn sub( &self, a: &Self::Element, b: &Self::Element, ) -> Result<Self::Element, ContextMismatchError>

Subtract two elements after checking their context.

fn mul( &self, a: &Self::Element, b: &Self::Element, ) -> Result<Self::Element, ContextMismatchError>

Multiply two elements after checking their context.

fn neg( &self, value: &Self::Element, ) -> Result<Self::Element, ContextMismatchError>

Negate an element after checking its context.

fn is_zero(&self, value: &Self::Element) -> bool

Test whether a value is the additive identity.

fn sum_products( &self, products: &[(&Self::Element, &Self::Element)], ) -> Result<Self::Element, ContextMismatchError>

Compute a sum of pairwise products in this context.

impl SemanticType for PadicRing

const TYPE_ID: &'static str = "noether/numthy/PadicRing@1"

Versioned, language-neutral semantic identity.

fn canonical_bytes(&self) -> Result<Vec<u8>, NoetherError>

Serialize without backend- or language-specific identity.

fn describe() -> &'static TypeDescription

Return static discovery metadata for the type.

impl Serialize for PadicRing

fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>

where S: Serializer,

Serialize this value into the given Serde serializer.

impl UnitInversion for PadicRing

fn inverse_unit( &self, value: &Self::Element, ) -> Result<Self::Element, UnitError>

Invert value when it belongs to this context and is a unit.

impl ValuedRing for PadicRing

fn uniformizer(&self) -> Self::Element

A chosen uniformizer.

fn valuation_lower_bound(&self, value: &Self::Element) -> Option<i64>

A proven lower bound for the valuation, or None for zero.

impl CommutativeRing for PadicRing

impl Eq for PadicRing