Struct Scheme 

pub struct Scheme<R: CommutativeRing> { /* private fields */ }

A closed affine or projective scheme with one canonical defining ideal.

Implementations

impl<R: Field + Display> Scheme<R>

pub fn base_change<M>( &self, map: &M, ) -> Result<Scheme<M::Target>, BaseChangeError>

where M: CoefficientMap<Source = R>, M::Target: Field,

Extend a closed affine or projective presentation along a checked field map, preserving its ambient kind and presentation data.

impl<R: Field> Scheme<R>

pub fn blow_up_at( &self, center: &SchemePoint<R>, ) -> Result<BlowUp<R>, GeometryError>

Construct verified affine charts for the blow-up at one rational point.

pub fn tangent_cone( &self, point: &SchemePoint<R>, ) -> Result<Scheme<R>, GeometryError>

Form the tangent cone from a degree-compatible Mora standard basis.

pub fn completion( &self, point: &SchemePoint<R>, truncation: SeriesTruncation, ) -> Result<SchemeCompletion<R>, GeometryError>

Complete at a rational point to one explicit finite monomial order.

impl Scheme<FiniteField>

pub fn point_counts<I>( &self, degrees: I, ) -> Result<PointCountSequence, PointCountError>

where I: IntoIterator<Item = u64>,

Count points over each requested relative extension degree.

impl<R: CommutativeRing> Scheme<R>

pub fn base(&self) -> &R

Base field context.

pub fn coordinate_ring(&self) -> &PolynomialRing<R>

Exact coordinate ring of the chosen ambient presentation.

pub fn defining_ideal(&self) -> &Ideal<PolynomialRing<R>>

Canonical defining ideal.

pub fn is_affine(&self) -> bool

Whether this presentation has an affine ambient space.

pub fn is_projective(&self) -> bool

Whether this presentation has a projective ambient space.

pub fn affine_ambient(&self) -> Option<&AffineSpace<R>>

Affine ambient space, when this is an affine presentation.

pub fn projective_ambient(&self) -> Option<&ProjectiveSpace<R>>

Projective ambient space, when this is a projective presentation.

impl Scheme<RationalField>

pub fn dimension(&self) -> Result<i64, SchemeError>

Dimension of this affine or projective presentation.

pub fn degree(&self) -> Result<Integer, SchemeError>

Standard-graded degree (multiplicity) of this presentation.

pub fn hilbert_polynomial( &self, ) -> Result<Polynomial<RationalField>, SchemeError>

Standard Hilbert polynomial of the homogeneous/filtered coordinate ring.

pub fn jacobian_subscheme(&self) -> Result<Self, SchemeError>

Rank-defect subscheme cut out by the expected-codimension Jacobian minors of the canonical defining basis.

pub fn singular_subscheme(&self) -> Result<Self, SchemeError>

Scheme-theoretic singular locus for an established complete- intersection presentation over this perfect field.

pub fn is_smooth(&self) -> Result<bool, SchemeError>

Decide smoothness when the exact complete-intersection Jacobian criterion applies.

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

Serialize this scheme presentation and canonical defining ideal.

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

Restore and revalidate this scheme presentation.

impl Scheme<FiniteField>

pub fn dimension(&self) -> Result<i64, SchemeError>

Dimension of this affine or projective presentation.

pub fn degree(&self) -> Result<Integer, SchemeError>

Standard-graded degree (multiplicity) of this presentation.

pub fn hilbert_polynomial( &self, ) -> Result<Polynomial<RationalField>, SchemeError>

Standard Hilbert polynomial of the homogeneous/filtered coordinate ring.

pub fn jacobian_subscheme(&self) -> Result<Self, SchemeError>

Rank-defect subscheme cut out by the expected-codimension Jacobian minors of the canonical defining basis.

pub fn singular_subscheme(&self) -> Result<Self, SchemeError>

Scheme-theoretic singular locus for an established complete- intersection presentation over this perfect field.

pub fn is_smooth(&self) -> Result<bool, SchemeError>

Decide smoothness when the exact complete-intersection Jacobian criterion applies.

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

Serialize this scheme presentation and canonical defining ideal.

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

Restore and revalidate this scheme presentation.

impl<R: Field> Scheme<R>

pub fn point<I>(&self, coordinates: I) -> Result<SchemePoint<R>, SchemeError>

where I: IntoIterator<Item = R::Element>,

Validate one rational point and normalize projective coordinates.

impl Scheme<FiniteField>

pub fn closed_points(&self) -> Result<FiniteSchemeClosedPointSet, SchemeError>

Solve a zero-dimensional affine scheme into closed points using a rational univariate representation and exact finite-field factorization.

This path never enumerates the base field. Proper residue extensions use deterministic Conway presentations when the current word- characteristic finite-field backend admits them.

pub fn rational_points(&self) -> Result<RationalPointEnumeration, SchemeError>

Stream all rational points in canonical order.

pub fn rational_points_shard( &self, shard_index: usize, shard_count: usize, ) -> Result<RationalPointEnumeration, SchemeError>

Stream one deterministic congruence shard of the canonical candidates.

pub fn point_count( &self, ) -> Result<(Integer, PointCountCertificate), SchemeError>

Count rational points with one deterministic compiled Rust pass.

pub fn point_count_partitioned( &self, shard_count: usize, ) -> Result<(Integer, PointCountCertificate), SchemeError>

Count rational points in deterministic checksum-bearing shards.

Trait Implementations

impl<R: Clone + CommutativeRing> Clone for Scheme<R>

fn clone(&self) -> Scheme<R>

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<R: CommutativeRing> Debug for Scheme<R>

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

Formats the value using the given formatter.

impl<'de> Deserialize<'de> for Scheme<FiniteField>

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

where D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl<'de> Deserialize<'de> for Scheme<RationalField>

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

where D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl<R: CommutativeRing> Display for Scheme<R>

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

Formats the value using the given formatter.

impl<R: PartialEq + CommutativeRing> PartialEq for Scheme<R>

fn eq(&self, other: &Scheme<R>) -> 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 SemanticType for Scheme<FiniteField>

const TYPE_ID: &'static str = "noether/commalg/Scheme@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 SemanticType for Scheme<RationalField>

const TYPE_ID: &'static str = "noether/commalg/Scheme@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<R> Serialize for Scheme<R>

where R: Field + Serialize, R::Element: Serialize,

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

where S: Serializer,

Serialize this value into the given Serde serializer.

impl<R: Eq + CommutativeRing> Eq for Scheme<R>

impl<R: CommutativeRing> StructuralPartialEq for Scheme<R>