Polynomials

This project implements a class template for handling rings of multivariate polynomials. The core component is the following class template:

template <typename T, std::size_t N = 0> class poly;

An object of this class represents a polynomial in a single variable: $$a_0 + a_1 x + a_2 x^2 + … + a_{N-1} x^{N-1}$$ where the coefficients $a_i$ are of type T. The value N represents the size of the polynomial (to avoid referring to it as "degree + 1"). The type T is required to form a ring, meaning it must support the binary operations +, -, *, as well as the unary -, and its default constructor should yield the additive identity (zero).

To extend the implementation to multivariate polynomials, the type T itself can be another poly. In general, a polynomial of m variables is represented recursively using the poly class up to depth m, with the innermost coefficient type no longer being a poly.

Compile with -std=c++20.

Example:

poly<poly<poly<int, 3>, 2>, 4>

represents a polynomial in three variables over the integer type. Conceptually, a polynomial in m variables is structured as follows:

• The outermost definition represents a polynomial in $x_1$, with coefficients being polynomials in variables $x_2$ through $x_m$.
• The coefficient type of this polynomial is itself a polynomial in $x_2$, with coefficients being polynomials in variables $x_3$ through $x_m$, and so on.

This recursive approach enables the representation and manipulation of polynomials with multiple variables in a structured and efficient manner.

Implemented Features

Constructors

• A default constructor initializes a polynomial identically equal to zero.
• Copy and move constructors allow initialization from another poly<U, M> where M ≤ N and U is convertible to T.
• A conversion constructor creates a polynomial of size 1 from a type convertible to T.
• A variadic constructor initializes the polynomial using up to N coefficients with perfect forwarding.
• The const_poly function constructs a polynomial of size 1 where the sole coefficient is another polynomial.
• Deduction guides enable template argument inference for poly objects.

Assignment Operators

• Copy and move assignment operators are implemented to support poly<U, M> where M ≤ N and U is convertible to T.

Arithmetic Operators

• Supports +=, -=, and *= with both polynomials and scalars.
• Implements +, -, * in binary form and unary -.
• Type deduction ensures the result has the minimal required size and a coefficient type of std::common_type_t<U, V>.

Indexing and Evaluation

• operator[](std::size_t i): Returns a reference to the coefficient $a_i$.
• at(...): Evaluates the polynomial as a function of multiple variables, supporting different argument counts and std::array<U, K> inputs.

Utility Methods

• size(): Returns the polynomial's size.

Cross Multiplication

• cross(p, q): multiplying two polynomials p and q of different variables, producing a new polynomial in the combined variable set.