# Dynamic Polynomials

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Sparse dynamic representation of multivariate polynomials that can be used with MultivariatePolynomials (see the documentation there for more information). Both commutative and non-commutative variables are supported. The following types are defined:

`Variable{V,M}`

: A variable which is commutative with`*`

when`V<:Commutative`

. Commutative variables are created using the`@polyvar`

macro, e.g.`@polyvar x y`

,`@polyvar x[1:8]`

and non-commutative variables are created likewise using the`@ncpolyvar`

macro. The type parameter`M`

is the monomial ordering.`Monomial{V,M}`

: A product of variables: e.g.`x*y^2`

.`MultivariatePolynomials.Term{T,Monomial{V,M}}`

: A product between an element of type`T`

and a`Monomial{V,M}`

, e.g`2x`

,`3.0x*y^2`

.`Polynomial{V,M,T}`

: A sum of`Term{T,Monomial{V,M}}`

, e.g.`2x + 3.0x*y^2 + y`

.

All common algebraic operations between those types are designed to be as efficient as possible without doing any assumption on `T`

.
Typically, one imagine `T`

to be a subtype of `Number`

but it can be anything.
This is useful for example in the package PolyJuMP where `T`

is often an affine expression of JuMP decision variables.
The commutativity of `T`

with `*`

is not assumed, even if it is the coefficient of a monomial of commutative variables.
However, commutativity of `T`

and of the variables `+`

is always assumed.
This allows to keep the terms sorted (Graded Lexicographic order is used) in polynomial and measure which enables more efficient operations.

Below is a simple usage example

julia> using DynamicPolynomials
julia> @polyvar x y # assigns x (resp. y) to a variable of name x (resp. y)
(x, y)
julia> p = 2x + 3.0x*y^2 + y # define a polynomial in variables x and y
y + 2.0x + 3.0xy²
julia> differentiate(p, x) # compute the derivative of p with respect to x
2.0 + 3.0y²
julia> differentiate.(p, (x, y)) # compute the gradient of p
(2.0 + 3.0y², 1.0 + 6.0xy)
julia> p((x, y)=>(y, x)) # replace any x by y and y by x
2.0y + x + 3.0x²y
julia> subs(p, y=>x^2) # replace any occurence of y by x^2
2.0x + x² + 3.0x⁵
julia> p(x=>1, y=>2) # evaluate p at [1, 2]
16.0

Below is an example with `@polyvar x[1:n]`

julia> n = 3;
julia> @polyvar x[1:n] # assign x to a tuple of variables x1, x2, x3
(Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, Graded{LexOrder}}[x₁, x₂, x₃],)
julia> p = sum(x .* x) # compute the sum of squares
x₃² + x₂² + x₁²
julia> subs(p, x[1]=>2, x[3]=>3) # make a partial substitution
13 + x₂²
julia> A = reshape(1:9, 3, 3);
julia> p(x => A * vec(x)) # corresponds to dot(A*x, A*x), need vec to convert the tuple to a vector
194x₃² + 244x₂x₃ + 77x₂² + 100x₁x₃ + 64x₁x₂ + 14x₁²

The terms of a polynomial are ordered in increasing monomial order. The default
ordering is the graded lex order but it can be modified using the
`monomial_order`

keyword argument of the `@polyvar`

macro.
We illustrate this below by borrowing the example p. 59 of "Ideals, Varieties and Algorithms"
of Cox, Little and O'Shea:

julia> p(x, y, z) = 4x*y^2*z + 4z^2 - 5x^3 + 7x^2*z^2
p (generic function with 1 method)
julia> @polyvar x y z monomial_order = LexOrder
(x, y, z)
julia> p(x, y, z)
4z² + 4xy²z + 7x²z² - 5x³
julia> @polyvar x y z
(x, y, z)
julia> p(x, y, z)
4z² - 5x³ + 4xy²z + 7x²z²
julia> @polyvar x y z monomial_order = Graded{Reverse{InverseLexOrder}}
(x, y, z)
julia> p(x, y, z)
4z² - 5x³ + 7x²z² + 4xy²z

Note that, when doing substitution, it is required to give the `Variable`

ordering that is meant.
Indeed, the ordering between the `Variable`

is not alphabetical but rather by order of creation
which can be undeterministic with parallel computing.
Therefore, this order cannot be used for substitution, even as a default (see here for a discussion about this).