ExtendedArithmetic.Polynomial
2023.288.635
dotnet add package ExtendedArithmetic.Polynomial version 2023.288.635
NuGet\InstallPackage ExtendedArithmetic.Polynomial Version 2023.288.635
<PackageReference Include="ExtendedArithmetic.Polynomial" Version="2023.288.635" />
paket add ExtendedArithmetic.Polynomial version 2023.288.635
#r "nuget: ExtendedArithmetic.Polynomial, 2023.288.635"
// Install ExtendedArithmetic.Polynomial as a Cake Addin
#addin nuget:?package=ExtendedArithmetic.Polynomial&version=2023.288.635
// Install ExtendedArithmetic.Polynomial as a Cake Tool
#tool nuget:?package=ExtendedArithmetic.Polynomial&version=2023.288.635
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. Please see the Other polynomial projects & numeric types heading below.
BigInteger Polynomial
 Supports symbolic univariate polynomial arithmetic including:
 Addition
 Subtraction
 Division
 Multiplication
 Modulus
 Factoring
 Derivatives
 Exponentiation
 GCD of polynomials
 Functional composition
 Irreducibility checking
 Polynomial evaluation by assigning values to the indeterminates
 Numeric values are of type BigInteger, so it support polynomials that evaluate to arbitrarily large numbers
 While all coefficients must be integers, it does support evaluating the polynomial with real and complex indeterminates, returning a real or complex result
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
Other polynomial projects and numeric types
I've written a number of other polynomial implementations and numeric types catering to various specific scenarios. Depending on what you're trying to do, another implementation of this same library might be more appropriate. All of my polynomial projects should have feature parity, where appropriate[^1].
[^1]: For example, the ComplexPolynomial implementation may be missing certain operations (namely: Irreducibility), because such a notion does not make sense or is ill defined in the context of complex numbers).
 Polynomial  The original. A univariate polynomial that uses System.Numerics.BigInteger as the indeterminate type.
 GenericPolynomial  A univariate polynomial library that allows the indeterminate to be of an arbitrary type, as long as said type implements operator overloading. This is implemented dynamically, at run time, calling the operator overload methods using Linq.Expressions and reflection.
 CSharp11Preview.GenericMath.Polynomial  A univariate polynomial library that allows the indeterminate to be of an arbitrary type, but this version is implemented using C# 11's new Generic Math via static virtual members in interfaces.
 GenericArithmetic  A core math library. Its a class of static methods that allows you to perform arithmetic on an arbitrary numeric type, represented by the generic type T, who's concrete type is decided by the caller. This is implemented using System.Linq.Expressions and reflection to resolve the type's static overloadable operator methods at runtime, so it works on all the .NET numeric types automagically, as well as any custom numeric type, provided it overloads the numeric operators and standard method names for other common functions (Min, Max, Abs, Sqrt, Parse, Sign, Log, Round, etc.). Every generic arithmetic class listed below takes a dependency on this class.
 MultivariatePolynomial  A multivariate polynomial (meaning more than one indeterminate, e.g. 2XY^2) which uses BigInteger as the type for the indeterminates.
 GenericMultivariatePolynomial  A multivariate polynomial that allows the indeterminates to be of [the same] arbitrary type. GenericMultivariatePolynomial is to MultivariatePolynomial what GenericPolynomial is to Polynomial, and indeed is implemented using the same strategy as GenericPolynomial (i.e. dynamic calling of the operator overload methods at runtime using Linq.Expressions and reflection).
 ComplexPolynomial  A univariate polynomial library that has System.Numerics.Complex type indeterminates.
 ComplexMultivariatePolynomial  A multivariate polynomial library that has System.Numerics.Complex indeterminates.
 BigDecimal  An arbitrary precision, base10 floating point number class.
 BigRational  Encodes a numeric value as an Integer + Fraction
 BigComplex  Essentially the same thing as System.Numerics.Complex but uses a System.Numerics.BigInteger type for the real and imaginary parts instead of a double.
 IntervalArithmetic. Instead of representing a value as a single number, interval arithmetic represents each value as a mathematical interval, or range of possibilities, [a,b], and allows the standard arithmetic operations to be performed upon them too, adjusting or scaling the underlying interval range as appropriate. See Wikipedia's article on Interval Arithmetic for further information.
 GNFS  A C# reference implementation of the General Number Field Sieve algorithm for the purpose of better understanding the General Number Field Sieve algorithm.
Product  Versions Compatible and additional computed target framework versions. 

.NET  net5.0 is compatible. net5.0windows was computed. net6.0 was computed. net6.0android was computed. net6.0ios was computed. net6.0maccatalyst was computed. net6.0macos was computed. net6.0tvos was computed. net6.0windows was computed. net7.0 was computed. net7.0android was computed. net7.0ios was computed. net7.0maccatalyst was computed. net7.0macos was computed. net7.0tvos was computed. net7.0windows was computed. net8.0 was computed. net8.0android was computed. net8.0ios was computed. net8.0maccatalyst was computed. net8.0macos was computed. net8.0tvos was computed. net8.0windows was computed. 
.NET Core  netcoreapp3.0 was computed. netcoreapp3.1 is compatible. 
.NET Standard  netstandard2.1 is compatible. 
.NET Framework  net45 is compatible. net451 was computed. net452 was computed. net46 is compatible. net461 was computed. net462 was computed. net463 was computed. net47 was computed. net471 was computed. net472 was computed. net48 is compatible. net481 was computed. 
MonoAndroid  monoandroid was computed. 
MonoMac  monomac was computed. 
MonoTouch  monotouch was computed. 
Tizen  tizen60 was computed. 
Xamarin.iOS  xamarinios was computed. 
Xamarin.Mac  xamarinmac was computed. 
Xamarin.TVOS  xamarintvos was computed. 
Xamarin.WatchOS  xamarinwatchos was computed. 

.NETCoreApp 3.1
 No dependencies.

.NETFramework 4.5
 No dependencies.

.NETFramework 4.6
 No dependencies.

.NETFramework 4.8
 No dependencies.

.NETStandard 2.1
 No dependencies.

net5.0
 No dependencies.
NuGet packages
This package is not used by any NuGet packages.
GitHub repositories
This package is not used by any popular GitHub repositories.
Version  Downloads  Last updated 

2023.288.635  158  10/15/2023 
2023.284.1620  107  10/11/2023 
2022.152.744  665  6/1/2022 
2021.364.1553  514  12/30/2021 
2021.364.1457  375  12/30/2021 
2021.234.2030  470  8/23/2021 
2.1.2021.45  475  2/14/2021 
2.1.0.1  500  1/9/2021 
2.1.0  586  10/12/2020 
2.0.0  647  10/2/2020 
1.2.0.19597  547  8/31/2020 
1.0.0.2  718  12/29/2019 
1.0.0.1  665  11/24/2019 
1.0.0  611  11/20/2019 