Class PolynomialRing<E>

java.lang.Object

org.episteme.core.mathematics.algebra.rings.PolynomialRing<E>

All Implemented Interfaces:

AbelianGroup<PolynomialRing.Polynomial<E>>, AbelianMonoid<PolynomialRing.Polynomial<E>>, Group<PolynomialRing.Polynomial<E>>, Magma<PolynomialRing.Polynomial<E>>, Monoid<PolynomialRing.Polynomial<E>>, Ring<PolynomialRing.Polynomial<E>>, Semiring<PolynomialRing.Polynomial<E>>, Set<PolynomialRing.Polynomial<E>>

public class PolynomialRing<E>
extends Object
implements Ring<PolynomialRing.Polynomial<E>>

Represents a Polynomial Ring R[x] over a ring R.

Elements are polynomials with coefficients in R.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	PolynomialRing.Polynomial<E>	Represents a polynomial P(x) = sum(a_i * x^i).

Constructor Summary

Constructors

Constructor	Description
PolynomialRing(Ring<E> coefficientRing, String variableName)

Method Summary

Modifier and Type	Method	Description
PolynomialRing.Polynomial<E>	add(PolynomialRing.Polynomial<E> a, PolynomialRing.Polynomial<E> b)	Returns the sum of two elements (additive notation).
boolean	contains(PolynomialRing.Polynomial<E> element)	Tests whether this set contains the specified element.
PolynomialRing.Polynomial<E>	create(Map<Integer,E> coefficients)
String	description()	Returns a human-readable description of this set.
Ring<E>	getCoefficientRing()
PolynomialRing.Polynomial<E>	inverse(PolynomialRing.Polynomial<E> element)	Returns the inverse of the given element.
boolean	isCommutative()	Abelian groups are always commutative by definition.
boolean	isEmpty()	Returns true if this set contains no elements.
boolean	isMultiplicationCommutative()	Tests whether multiplication is commutative in this semiring.
PolynomialRing.Polynomial<E>	multiply(PolynomialRing.Polynomial<E> a, PolynomialRing.Polynomial<E> b)	Returns the product of two elements.
PolynomialRing.Polynomial<E>	negate(PolynomialRing.Polynomial<E> element)	Returns the additive inverse (negation) of an element.
PolynomialRing.Polynomial<E>	one()	Returns the multiplicative identity (one element).
PolynomialRing.Polynomial<E>	operate(PolynomialRing.Polynomial<E> left, PolynomialRing.Polynomial<E> right)	Performs the binary operation on two elements.
PolynomialRing.Polynomial<E>	zero()	Returns the additive identity (zero element).

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface AbelianGroup

subtract

Methods inherited from interface AbelianMonoid

identity

Methods inherited from interface Monoid

isAssociative

Methods inherited from interface Ring

hasUnity

Methods inherited from interface Semiring

pow

Constructor Details

PolynomialRing

public PolynomialRing(Ring<E> coefficientRing,
 String variableName)

Method Details

getCoefficientRing

public Ring<E> getCoefficientRing()

operate

public PolynomialRing.Polynomial<E> operate(PolynomialRing.Polynomial<E> left,
 PolynomialRing.Polynomial<E> right)

Description copied from interface: Magma

Performs the binary operation on two elements.

This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

Properties: None required (not necessarily associative or commutative)

Specified by:

operate in interface Magma<E>

Parameters:

left - the first operand

right - the second operand

Returns:

the result of a ∗ b

See Also:

• Magma.isAssociative()
• Magma.isCommutative()

add

public PolynomialRing.Polynomial<E> add(PolynomialRing.Polynomial<E> a,
 PolynomialRing.Polynomial<E> b)

Description copied from interface: AbelianMonoid

Returns the sum of two elements (additive notation).

Delegates to Magma.operate(Object, Object).

Specified by:

add in interface AbelianMonoid<E>

Parameters:

a - the first addend

b - the second addend

Returns:

a + b

zero

public PolynomialRing.Polynomial<E> zero()

Description copied from interface: AbelianMonoid

Returns the additive identity (zero element).

Delegates to AbelianMonoid.identity().

Specified by:

zero in interface AbelianMonoid<E>

Returns:

the zero element

negate

public PolynomialRing.Polynomial<E> negate(PolynomialRing.Polynomial<E> element)

Description copied from interface: AbelianGroup

Returns the additive inverse (negation) of an element.

Satisfies: a + (-a) = (-a) + a = 0

Specified by:

negate in interface AbelianGroup<E>

Parameters:

element - the element to negate

Returns:

-element

See Also:

• Group.inverse(Object)

inverse

public PolynomialRing.Polynomial<E> inverse(PolynomialRing.Polynomial<E> element)

Description copied from interface: Group

Returns the inverse of the given element.

For element a, returns a⁻¹ such that: a ∗ a⁻¹ = a⁻¹ ∗ a = e (identity).

Specified by:

inverse in interface Group<E>

Parameters:

element - the element to invert

Returns:

the inverse element

multiply

public PolynomialRing.Polynomial<E> multiply(PolynomialRing.Polynomial<E> a,
 PolynomialRing.Polynomial<E> b)

Description copied from interface: Semiring

Returns the product of two elements.

Multiplication must be associative and distribute over addition.

Specified by:

multiply in interface Semiring<E>

Parameters:

a - the first factor

b - the second factor

Returns:

a × b

one

public PolynomialRing.Polynomial<E> one()

Description copied from interface: Semiring

Returns the multiplicative identity (one element).

Satisfies: 1 × a = a × 1 = a for all elements a.

Specified by:

one in interface Semiring<E>

Returns:

the multiplicative identity

isCommutative

public boolean isCommutative()

Description copied from interface: AbelianGroup

Abelian groups are always commutative by definition.

Specified by:

isCommutative in interface AbelianGroup<E>

Specified by:

isCommutative in interface AbelianMonoid<E>

Specified by:

isCommutative in interface Group<E>

Specified by:

isCommutative in interface Magma<E>

Returns:

always true

isMultiplicationCommutative

public boolean isMultiplicationCommutative()

Description copied from interface: Semiring

Tests whether multiplication is commutative in this semiring.

Specified by:

isMultiplicationCommutative in interface Semiring<E>

Returns:

true if multiplication commutes

description

public String description()

Description copied from interface: Set

Returns a human-readable description of this set.

Examples:

• "ℝ (Real Numbers)"
• "ℤ/12ℤ (Integers modulo 12)"
• "{1, 2, 3, 4, 5}"

Specified by:

description in interface Set<E>

Returns:

a description of this set

isEmpty

public boolean isEmpty()

Description copied from interface: Set

Returns true if this set contains no elements.

The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

Specified by:

isEmpty in interface Set<E>

Returns:

true if this set is empty

contains

public boolean contains(PolynomialRing.Polynomial<E> element)

Description copied from interface: Set

Tests whether this set contains the specified element.

This is the fundamental operation of a set - membership testing.

Specified by:

contains in interface Set<E>

Parameters:

element - the element to test for membership

Returns:

true if this set contains the element, false otherwise

See Also:

• Set.isEmpty()
• invalid reference

  #size()

create

public PolynomialRing.Polynomial<E> create(Map<Integer,E> coefficients)