ExtendedArithmetic.Polynomial
2.1.0.1
A univariate, sparse, symbolic, BigInteger polynomial arithmetic class.
InstallPackage ExtendedArithmetic.Polynomial Version 2.1.0.1
dotnet add package ExtendedArithmetic.Polynomial version 2.1.0.1
<PackageReference Include="ExtendedArithmetic.Polynomial" Version="2.1.0.1" />
paket add ExtendedArithmetic.Polynomial version 2.1.0.1
Polynomial
A univariate, sparse, integer polynomial class. That is, a polynomial in only one indeterminate, X, that only tracks terms with nonzero coefficients, and all coefficients are BigInteger integers.
NOTE: All arithmetic is done symbolically. That means the result a arithmetic operation on two polynomials, returns another polynomial, not some integer that is the result of evaluating said polynomials.
Generic Arithmetic Types
 I created an implementation that can perform symbolic polynomial arithmetic on generic numeric types. All polynomial arithmetic is performed on this generic type, allowing BigInteger to be swapped out for Complex, Decimal, Double, BigComplex, BigDecimal, BigRational, Int32, Int64 and more! Check it out on my GenericArithmeticExpression branch.
BigInteger Polynomial
 Supports symbolic univariate polynomial arithmetic including:
 Addition
 Subtraction
 Multiplication
 Division
 Modulus
 Exponentiation
 GCD of polynomials
 Irreducibility checking
 Polynomial evaluation by assigning to the invariant (X in this case) a value.
 All numbers use BigInteger integers, for arbitrarily large numbers.
Polynomial Rings over a Finite Field

Polynomial.Field supports all of the above arithmetic operations, but on a polynomial ring over a finite field!
 What this effectively means in lesstechnical terms is that the polynomial arithmetic is performed in the usual way, but the result is then taken modulus two things: A BigInteger integer and another polynomial:
 Modulus an integer: All the polynomial coefficients are reduced modulus this integer.
 Modulus a polynomial: The whole polynomial is reduced modulus another, smaller, polynomial. This notion works much the same as regular modulus; The modulus polynomial, lets call it g, is declared to be zero, and so every multiple of g is reduced to zero. You can think of it this way (although this is not how its actually carried out): From a large polynomial, g is repeatedly subtracted from that polynomial until it cant subtract g anymore without going past zero. The result is a polynomial that lies between 0 and g. Just like regular modulus, the result is always less than your modulus, or zero if the first polynomial was an exact multiple of the modulus.
 Effectively forms a quotient ring
 What this effectively means in lesstechnical terms is that the polynomial arithmetic is performed in the usual way, but the result is then taken modulus two things: A BigInteger integer and another polynomial:

You can instantiate a polynomial in various ways:
 From a string
 This is the most massivelyuseful way and is the quickest way to start working with a particular polynomial you had in mind.
 From its roots
 Build a polynomial that has as its roots, all of the numbers in the supplied array. If you want multiplicity of roots, include that number in the array multiple times.
 From the basem expansion of a number
 Given a large number and a radix (base), call it m, a polynomial will be generated that is that number represented in the number base m.
 From a string

Other methods of interest that are related to, but not necessarily performed on a polynomial:
 Eulers Criterion
 Legendre Symbol and Legendre Symbol Search
 TonelliShanks
 Chinese Remainder Theorem
Polynomial
A univariate, sparse, integer polynomial class. That is, a polynomial in only one indeterminate, X, that only tracks terms with nonzero coefficients, and all coefficients are BigInteger integers.
NOTE: All arithmetic is done symbolically. That means the result a arithmetic operation on two polynomials, returns another polynomial, not some integer that is the result of evaluating said polynomials.
Generic Arithmetic Types
 I created an implementation that can perform symbolic polynomial arithmetic on generic numeric types. All polynomial arithmetic is performed on this generic type, allowing BigInteger to be swapped out for Complex, Decimal, Double, BigComplex, BigDecimal, BigRational, Int32, Int64 and more! Check it out on my GenericArithmeticExpression branch.
BigInteger Polynomial
 Supports symbolic univariate polynomial arithmetic including:
 Addition
 Subtraction
 Multiplication
 Division
 Modulus
 Exponentiation
 GCD of polynomials
 Irreducibility checking
 Polynomial evaluation by assigning to the invariant (X in this case) a value.
 All numbers use BigInteger integers, for arbitrarily large numbers.
Polynomial Rings over a Finite Field

Polynomial.Field supports all of the above arithmetic operations, but on a polynomial ring over a finite field!
 What this effectively means in lesstechnical terms is that the polynomial arithmetic is performed in the usual way, but the result is then taken modulus two things: A BigInteger integer and another polynomial:
 Modulus an integer: All the polynomial coefficients are reduced modulus this integer.
 Modulus a polynomial: The whole polynomial is reduced modulus another, smaller, polynomial. This notion works much the same as regular modulus; The modulus polynomial, lets call it g, is declared to be zero, and so every multiple of g is reduced to zero. You can think of it this way (although this is not how its actually carried out): From a large polynomial, g is repeatedly subtracted from that polynomial until it cant subtract g anymore without going past zero. The result is a polynomial that lies between 0 and g. Just like regular modulus, the result is always less than your modulus, or zero if the first polynomial was an exact multiple of the modulus.
 Effectively forms a quotient ring
 What this effectively means in lesstechnical terms is that the polynomial arithmetic is performed in the usual way, but the result is then taken modulus two things: A BigInteger integer and another polynomial:

You can instantiate a polynomial in various ways:
 From a string
 This is the most massivelyuseful way and is the quickest way to start working with a particular polynomial you had in mind.
 From its roots
 Build a polynomial that has as its roots, all of the numbers in the supplied array. If you want multiplicity of roots, include that number in the array multiple times.
 From the basem expansion of a number
 Given a large number and a radix (base), call it m, a polynomial will be generated that is that number represented in the number base m.
 From a string

Other methods of interest that are related to, but not necessarily performed on a polynomial:
 Eulers Criterion
 Legendre Symbol and Legendre Symbol Search
 TonelliShanks
 Chinese Remainder Theorem
Dependencies
This package has no dependencies.
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