Types

AbstractVariable <: AbstractMonomialLike

Abstract type for a variable.

variable(p::AbstractPolynomialLike)

Converts p to a variable. Throws InexactError if it is not possible.

Examples

Calling variable(x^2 + x - x^2) should return the variable x and calling variable(1.0y) should return the variable y however calling variable(2x) or variable(x + y) should throw InexactError.

Note

This operation is not type stable for the TypedPolynomials implementation if nvariables(p) > 1 but is type stable for DynamicPolynomials.

name(v::AbstractVariable)::AbstractString

Returns the name of a variable.

name_base_indices(v::AbstractVariable)

Returns the name of the variable (as a String or Symbol) and its indices as a Vector{Int} or tuple of Ints.

variable_union_type(p::AbstractPolynomialLike)

Return the supertype for variables of p. If p is a variable, it should not be the type of p but the supertype of all variables that could be created.

Examples

For TypedPolynomials, a variable of name x has type Variable{:x} so variable_union_type should return Variable. For DynamicPolynomials, all variables have the same type Variable{C} where C is true for commutative variables and false for non-commutative ones so variable_union_type should return Variable{C}.

similar_variable(p::AbstractPolynomialLike, variable::Type{Val{V}})

Creates a new variable V based upon the the given source polynomial.

similar_variable(p::AbstractPolynomialLike, v::Symbol)

Creates a new variable based upon the given source polynomial and the given symbol v. Note that this can lead to type instabilities.

Examples

Calling similar_variable(typedpoly, Val{:x}) on a polynomial created with TypedPolynomials results in TypedPolynomials.Variable{:x}.

@similar_variable(p::AbstractPolynomialLike, variable)

Calls similar_variable(p, Val{variable}) and binds the result to a variable with the same name.

Examples

Calling @similar_variable typedpoly x on a polynomial created with TypedPolynomials binds TypedPolynomials.Variable{:x} to the variable x.

conj(x::AbstractVariable)

Return the complex conjugate of a given variable if it was declared as a complex variable; else return the variable unchanged.

See also isreal, isconj.

real(x::AbstractVariable)

Return the real part of a given variable if it was declared as a complex variable; else return the variable unchanged.

See also imag.

imag(x::AbstractVariable)

Return the imaginary part of a given variable if it was declared as a complex variable; else return zero.

See also isreal, isimagpart, real.

isreal(x::AbstractVariable)

Return whether a given variable was declared as a real-valued or complex-valued variable (also their conjugates are complex, but their real and imaginary parts are not). By default, all variables are real-valued.

isrealpart(x::AbstractVariable)

Return whether the given variable is the real part of a complex-valued variable.

See also isreal, isimagpart, isconj.

isimagpart(x::AbstractVariable)

Return whether the given variable is the imaginary part of a complex-valued variable.

See also isreal, isrealpart, isconj.

isconj(x::AbstractVariable)

Return whether the given variable is obtained by conjugating a user-defined complex-valued variable.

See also isreal, isrealpart, isimagpart.

ordinary_variable(x::Union{AbstractVariable, AbstractVector{<:AbstractVariable}})

Given some (complex-valued) variable that was transformed by conjugation, taking its real part, or taking its imaginary part, return the original variable as it was defined by the user.

See also conj, real, imag.

AbstractMonomialLike

Abstract type for a value that can act like a monomial. For instance, an AbstractVariable is an AbstractMonomialLike since it can act as a monomial of one variable with degree 1.

AbstractMonomial <: AbstractMonomialLike

Abstract type for a monomial, i.e. a product of variables elevated to a nonnegative integer power.

monomial_type(p::AbstractPolynomialLike)

Return the type of the monomials of p.

term_type(::Type{PT}) where PT<:AbstractPolynomialLike

Returns the type of the monomials of a polynomial of type PT.

variables(p::AbstractPolynomialLike)

Returns the tuple of the variables of p in decreasing order. It could contain variables of zero degree, see the example section.

Examples

Calling variables(x^2*y) should return (x, y) and calling variables(x) should return (x,). Note that the variables of m does not necessarily have nonzero degree. For instance, variables([x^2*y, y*z][1]) is usually (x, y, z) since the two monomials have been promoted to a common type.

effective_variables(p::AbstractPolynomialLike)

Return a vector of eltype variable_union_type(p) (see variable_union_type), containing all the variables that has nonzero degree in at least one term. That is, return all the variables v such that maxdegree(p, v) is not zero. The returned vector is sorted in decreasing order.

nvariables(p::AbstractPolynomialLike)

Returns the number of variables in p, i.e. length(variables(p)). It could be more than the number of variables with nonzero degree (see the Examples section of variables).

Examples

Calling nvariables(x^2*y) should return at least 2 and calling nvariables(x) should return at least 1.

exponents(t::AbstractTermLike)

Returns the exponent of the variables in the monomial of the term t.

Examples

Calling exponents(x^2*y) should return (2, 1).

degree(t::AbstractTermLike)

Returns the total degree of the monomial of the term t, i.e. sum(exponents(t)).

degree(t::AbstractTermLike, v::AbstractVariable)

Returns the exponent of the variable v in the monomial of the term t.

Examples

Calling degree(x^2*y) should return 3 which is $2 + 1$. Calling degree(x^2*y, x) should return 2 and calling degree(x^2*y, y) should return 1.

isconstant(t::AbstractTermLike)

Returns whether the monomial of t is constant.

powers(t::AbstractTermLike)

Returns an iterator over the powers of the monomial of t.

Examples

Calling powers(3x^4*y) should return((x, 4), (y, 1))`.

constant_monomial(p::AbstractPolynomialLike)

Returns a constant monomial of the monomial type of p with the same variables as p.

constant_monomial(::Type{PT}) where {PT<:AbstractPolynomialLike}

Returns a constant monomial of the monomial type of a polynomial of type PT.

map_exponents(f, m1::AbstractMonomialLike, m2::AbstractMonomialLike)

If $m_1 = \prod x^{\alpha_i}$ and $m_2 = \prod x^{\beta_i}$ then it returns the monomial $m = \prod x^{f(\alpha_i, \beta_i)}$.

Examples

The multiplication m1 * m2 is equivalent to map_exponents(+, m1, m2), the unsafe division div_multiple(m1, m2) is equivalent to map_exponents(-, m1, m2), gcd(m1, m2) is equivalent to map_exponents(min, m1, m2), lcm(m1, m2) is equivalent to map_exponents(max, m1, m2).

multiplication_preserves_monomial_order(P::Type{<:AbstractPolynomialLike})

Returns a Bool indicating whether the order is preserved in the multiplication of monomials of type monomial_type(P). That is, if a < b then a * c < b * c for any monomial c. This returns true by default. This is used by Polynomial so a monomial type for which the multiplication does not preserve the monomial order can still be used with Polynomial if it implements a method for this function that returns false.

abstract type AbstractMonomialOrdering end

Abstract type for monomial ordering as defined in [CLO13, Definition 2.2.1, p. 55]

[CLO13] Cox, D., Little, J., & OShea, D. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013.

ordering(p::AbstractPolynomialLike)

Returns the AbstractMonomialOrdering used for the monomials of p.

compare(a, b, order::Type{<:AbstractMonomialOrdering})

Returns a negative number if a < b, a positive number if a > b and zero if a == b. The comparison is done according to order.

struct LexOrder <: AbstractMonomialOrdering end

Lexicographic (Lex for short) Order often abbreviated as lex order as defined in [CLO13, Definition 2.2.3, p. 56]

The Graded version is often abbreviated as grlex order and is defined in [CLO13, Definition 2.2.5, p. 58]

[CLO13] Cox, D., Little, J., & OShea, D. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013.

struct InverseLexOrder <: AbstractMonomialOrdering end

Inverse Lex Order defined in [CLO13, Exercise 2.2.6, p. 61] where it is abbreviated as invlex. It corresponds to LexOrder but with the variables in reverse order.

The Graded version can be abbreviated as grinvlex order. It is defined in [BDD13, Definition 2.1] where it is called Graded xel order.

[CLO13] Cox, D., Little, J., & OShea, D. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013. [BDD13] Batselier, K., Dreesen, P., & De Moor, B. The geometry of multivariate polynomial division and elimination. SIAM Journal on Matrix Analysis and Applications, 34(1), 102-125, 2013.

struct Graded{O<:AbstractMonomialOrdering} <: AbstractMonomialOrdering
    same_degree_ordering::O
end

Monomial ordering defined by:

• degree(a) == degree(b) then the ordering is determined by same_degree_ordering,
• otherwise, it is the ordering between the integers degree(a) and degree(b).

struct Reverse{O<:AbstractMonomialOrdering} <: AbstractMonomialOrdering
    reverse_order::O
end

Monomial ordering defined by compare(a, b, ::Type{Reverse{O}}) where {O} = compare(b, a, O).

Reverse Lex Order defined in [CLO13, Exercise 2.2.9, p. 61] where it is abbreviated as rinvlex. can be obtained as Reverse{InverseLexOrder}.

The Graded Reverse Lex Order often abbreviated as grevlex order defined in [CLO13, Definition 2.2.6, p. 58] can be obtained as Graded{Reverse{InverseLexOrder}}.

[CLO13] Cox, D., Little, J., & OShea, D. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013.

AbstractTermLike{T}

Abstract type for a value that can act like a term. For instance, an AbstractMonomial is an AbstractTermLike{Int} since it can act as a term with coefficient 1.

AbstractTerm{T} <: AbstractTermLike{T}

Abstract type for a term of coefficient type T, i.e. the product between a value of type T and a monomial.

struct Term{CoeffType,M<:AbstractMonomial} <: AbstractTerm{CoeffType}
    coefficient::CoeffType
    monomial::M
end

A representation of the multiplication between a coefficient and a monomial.

term(coef, mono::AbstractMonomialLike)

Returns a term with coefficient coef and monomial mono. There are two key difference between this and coef * mono:

• term(coef, mono) does not copy coef and mono so modifying this term with MutableArithmetics may modifying the input of this function. To avoid this, call term(MA.copy_if_mutable(coef), MA.copy_if_mutable(mono)) where MA = MutableArithmetics.

• Suppose that coef = (x + 1) and mono = x^2, coef * mono gives the polynomial with integer coefficients x^3 + x^2 which term(x + 1, x^2) gives a term with polynomial coefficient x + 1.

  term(p::AbstractPolynomialLike)

Converts the polynomial p to a term. When applied on a polynomial, it throws an InexactError if it has more than one term. When applied to a term, it is the identity and does not copy it. When applied to a monomial, it create a term of type AbstractTerm{Int}.

term_type(p::AbstractPolynomialLike)

Returns the type of the terms of p.

term_type(::Type{PT}) where PT<:AbstractPolynomialLike

Returns the type of the terms of a polynomial of type PT.

term_type(p::AbstractPolynomialLike, ::Type{T}) where T

Returns the type of the terms of p but with coefficient type T.

term_type(::Type{PT}, ::Type{T}) where {PT<:AbstractPolynomialLike, T}

Returns the type of the terms of a polynomial of type PT but with coefficient type T.

coefficient(t::AbstractTermLike)

Returns the coefficient of the term t.

coefficient(p::AbstractPolynomialLike, m::AbstractMonomialLike)

Returns the coefficient of the monomial m in p.

Examples

Calling coefficient on $4x^2y$ should return $4$. Calling coefficient(2x + 4y^2 + 3, y^2) should return $4$. Calling coefficient(2x + 4y^2 + 3, x^2) should return $0$.

coefficient_type(p::AbstractPolynomialLike)

Returns the coefficient type of p.

coefficient_type(::Type{PT}) where PT

Returns the coefficient type of a polynomial of type PT.

Examples

Calling coefficient_type on $(4//5)x^2y$ should return Rational{Int}, calling coefficient_type on $1.0x^2y + 2.0x$ should return Float64 and calling coefficient_type on $xy$ should return Int.

monomial(t::AbstractTermLike)

Returns the monomial of the term t.

Examples

Calling monomial on $4x^2y$ should return $x^2y$.

constant_term(α, p::AbstractPolynomialLike)

Creates a constant term with coefficient α and the same variables as p.

constant_term(α, ::Type{PT} where {PT<:AbstractPolynomialLike}

Creates a constant term of the term type of a polynomial of type PT.

zero_term(p::AbstractPolynomialLike{T}) where T

Equivalent to constant_term(zero(T), p).

zero_term(α, ::Type{PT} where {T, PT<:AbstractPolynomialLike{T}}

Equivalent to constant_term(zero(T), PT).

degree_complex(t::AbstractTermLike)

Return the total complex degree of the monomial of the term t, i.e., the maximum of the total degree of the declared variables in t and the total degree of the conjugate variables in t. To be well-defined, the monomial must not contain real parts or imaginary parts of variables. If x₁ and x₂ are real-valued variables and z₁ and z₂ are complex-valued,

• degree_complex(x₁^2 * x₂^3) = 5
• degree_complex(z₁^3 * conj(z₁)^4) = max(3, 4) = 4 and degree_complex(z₁^4 * conj(z₁)^3) = max(4, 3) = 4
• degree_complex(z₁^3 * z₂ * conj(z₁)^2 * conj(z₂)^4) = max(4, 6) = 6 and degree_complex(z₁^4 * z₂ * conj(z₁) * conj(z₂)^3) = max(5, 4) = 5
• degree_complex(x₁^2 * x₂^3 * z₁^3 * z₂ * conj(z₁)^2 * conj(z₂)^4) = 5 + max(4, 6) = 11

degree_complex(t::AbstractTermLike, v::AbstractVariable)

Returns the exponent of the variable v or its conjugate in the monomial of the term t, whatever is larger.

See also isconj.

halfdegree(t::AbstractTermLike)

Return the equivalent of ceil(degree(t)/2)for real-valued terms ordegree_complex(t)for terms with only complex variables; however, respect any mixing between complex and real-valued variables. To be well-defined, the monomial must not contain real parts or imaginary parts of variables. Ifx₁andx₂are real-valued variables andz₁andz₂` are complex-valued,

• halfdegree(x₁^2 * x₂^3) = ⌈5/2⌉ = 3
• halfdegree(z₁^3 * conj(z₁)^4) = max(3, 4) = 4 and halfdegree(z₁^4 * conj(z₁)^3) = max(4, 3) = 4
• halfdegree(z₁^3 * z₂ * conj(z₁)^2 * conj(z₂)^4) = max(4, 6) = 6 and halfdegree(z₁^4 * z₂ * conj(z₁) * conj(z₂)^3) = max(5, 4) = 5
• halfdegree(x₁^2 * x₂^3 * z₁^3 * z₂ * conj(z₁)^2 * conj(z₂)^4) = ⌈5/2⌉ + max(4, 6) = 9

AbstractPolynomialLike{T}

Abstract type for a value that can act like a polynomial. For instance, an AbstractTerm{T} is an AbstractPolynomialLike{T} since it can act as a polynomial of only one term.

AbstractPolynomial{T} <: AbstractPolynomialLike{T}

Abstract type for a polynomial of coefficient type T, i.e. a sum of AbstractTerm{T}s.

struct Polynomial{CoeffType,T<:AbstractTerm{CoeffType},V<:AbstractVector{T}} <: AbstractPolynomial{CoeffType}
    terms::V
end

Representation of a multivariate polynomial as a vector of nonzero terms sorted in ascending monomial order.

polynomial(p::AbstractPolynomialLike)

Converts p to a value with polynomial type.

polynomial(p::AbstractPolynomialLike, ::Type{T}) where T

Converts p to a value with polynomial type with coefficient type T.

polynomial(a::AbstractVector, mv::AbstractVector{<:AbstractMonomialLike})

Creates a polynomial equal to dot(a, mv).

polynomial(terms::AbstractVector{<:AbstractTerm}, s::ListState=MessyState())

Creates a polynomial equal to sum(terms) where terms are guaranteed to be in state s.

polynomial(f::Function, mv::AbstractVector{<:AbstractMonomialLike})

Creates a polynomial equal to sum(f(i) * mv[i] for i in 1:length(mv)).

Examples

Calling polynomial([2, 4, 1], [x, x^2*y, x*y]) should return $4x^2y + xy + 2x$.

polynomial_type(p::AbstractPolynomialLike)

Returns the type that p would have if it was converted into a polynomial.

polynomial_type(::Type{PT}) where PT<:AbstractPolynomialLike

Returns the same as polynomial_type(::PT).

polynomial_type(p::AbstractPolynomialLike, ::Type{T}) where T

Returns the type that p would have if it was converted into a polynomial of coefficient type T.

polynomial_type(::Type{PT}, ::Type{T}) where {PT<:AbstractPolynomialLike, T}

Returns the same as polynomial_type(::PT, ::Type{T}).

terms(p::AbstractPolynomialLike)

Returns an iterator over the nonzero terms of the polynomial p sorted in the decreasing monomial order.

Examples

Calling terms on $4x^2y + xy + 2x$ should return an iterator of $[4x^2y, xy, 2x]$.

nterms(p::AbstractPolynomialLike)

Returns the number of nonzero terms in p, i.e. length(terms(p)).

Examples

Calling nterms on $4x^2y + xy + 2x$ should return 3.

coefficients(p::AbstractPolynomialLike)

Returns an iterator over the coefficients of p of the nonzero terms of the polynomial sorted in the decreasing monomial order.

coefficients(p::AbstractPolynomialLike, X::AbstractVector)

Returns an iterator over the coefficients of the monomials of X in p where X is a monomial vector not necessarily sorted but with no duplicate entry.

Examples

Calling coefficients on $4x^2y + xy + 2x$ should return an iterator of $[4, 1, 2]$. Calling coefficients(4x^2*y + x*y + 2x + 3, [x, 1, x*y, y]) should return an iterator of $[2, 3, 1, 0]$.

coefficient(p::AbstractPolynomialLike, m::AbstractMonomialLike, vars)::AbstractPolynomialLike

Returns the coefficient of the monomial m of the polynomial p considered as a polynomial in variables vars.

Example

Calling coefficient((a+b)x^2+2x+y*x^2, x^2, [x,y]) should return a+b. Calling coefficient((a+b)x^2+2x+y*x^2, x^2, [x]) should return a+b+y.

monomials(p::AbstractPolynomialLike)

Returns an iterator over the monomials of p of the nonzero terms of the polynomial sorted in the decreasing order.

monomials(vars::Tuple, degs::AbstractVector{Int}, filter::Function = m -> true)

Builds the vector of all the monomial_vector m with variables vars such that the degree degree(m) is in degs and filter(m) is true.

Examples

Calling monomials on $4x^2y + xy + 2x$ should return an iterator of $[x^2y, xy, x]$.

Calling monomials((x, y), [1, 3], m -> degree(m, y) != 1) should return [x^3, x*y^2, y^3, x] where x^2*y and y have been excluded by the filter.

mindegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractPolynomialLike}})

Returns the minimal total degree of the monomials of p, i.e. minimum(degree, terms(p)).

mindegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractPolynomialLike}}, v::AbstractVariable)

Returns the minimal degree of the monomials of p in the variable v, i.e. minimum(degree.(terms(p), v)).

Examples

Calling mindegree on on $4x^2y + xy + 2x$ should return 1, mindegree(4x^2y + xy + 2x, x) should return 1 and mindegree(4x^2y + xy + 2x, y) should return 0.

maxdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Returns the maximal total degree of the monomials of p, i.e. maximum(degree, terms(p)).

maxdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}}, v::AbstractVariable)

Returns the maximal degree of the monomials of p in the variable v, i.e. maximum(degree.(terms(p), v)).

Examples

Calling maxdegree on $4x^2y + xy + 2x$ should return 3, maxdegree(4x^2y + xy + 2x, x) should return 2 and maxdegree(4x^2y + xy + 2x, y) should return 1.

extdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractPolynomialLike}})

Returns the extremal total degrees of the monomials of p, i.e. (mindegree(p), maxdegree(p)).

extdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractPolynomialLike}}, v::AbstractVariable)

Returns the extremal degrees of the monomials of p in the variable v, i.e. (mindegree(p, v), maxdegree(p, v)).

Examples

Calling extdegree on $4x^2y + xy + 2x$ should return (1, 3), extdegree(4x^2y + xy + 2x, x) should return (1, 2) and maxdegree(4x^2y + xy + 2x, y) should return (0, 1).

leading_term(p::AbstractPolynomialLike)

Returns the leading term, i.e. last(terms(p)).

Examples

Calling leading_term on $4x^2y + xy + 2x$ should return $4x^2y$.

leading_coefficient(p::AbstractPolynomialLike)

Returns the coefficient of the leading term of p, i.e. coefficient(leading_term(p)).

Examples

Calling leading_coefficient on $4x^2y + xy + 2x$ should return $4$ and calling it on $0$ should return $0$.

leading_monomial(p::AbstractPolynomialLike)

Returns the monomial of the leading term of p, i.e. monomial(leading_term(p)) or last(monomials(p)).

Examples

Calling leading_monomial on $4x^2y + xy + 2x$ should return $x^2y$.

deg_num_leading_terms(p::AbstractPolynomialLike, var)

Return deg, num where deg = maxdegree(p, var) and num is the number of terms t such that degree(t, var) == deg.

remove_leading_term(p::AbstractPolynomialLike)

Returns a polynomial with the leading term removed in the polynomial p.

Examples

Calling remove_leading_term on $4x^2y + xy + 2x$ should return $xy + 2x$.

Returns a polynomial with the terms having their monomial in the monomial vector mv removed in the polynomial p.

Examples

Calling remove_monomials(4x^2*y + x*y + 2x, [x*y]) should return $4x^2*y + 2x$.

function filter_terms(f::Function, p::AbstractPolynomialLike)

Filter the polynomial p by only keep the terms t such that f(p) is true.

See also OfDegree.

Examples

julia> p = 1 - 2x + x * y - 3y^2 + x^2 * y
1 - 2x - 3y² + xy + x²y

julia> filter_terms(OfDegree(2), p)
-3y² + xy

julia> filter_terms(!OfDegree(2), p)
1 - 2x + x²y

julia> filter_terms(!OfDegree(0:2), p)
x²y

julia> filter_terms(iseven ∘ coefficient, p)
-2x

struct OfDegree{D} <: Function
    degree::D
end

A function d::OfDegree is such that d(t) returns degree(t) == d.degree. Note that !d creates the negation. See also filter_terms.

monic(p::AbstractPolynomialLike)

Returns p / leading_coefficient(p) where the leading coefficient of the returned polynomials is made sure to be exactly one to avoid rounding error.

map_coefficients(f::Function, p::AbstractPolynomialLike, nonzero = false)

Returns a polynomial with the same monomials as p but each coefficient α is replaced by f(α). The function may return zero in which case the term is dropped. If the function is known to never return zero for a nonzero input, nonzero can be set to true to get a small speedup.

See also map_coefficients! and map_coefficients_to!.

Examples

Calling map_coefficients(α -> mod(3α, 6), 2x*y + 3x + 1) should return 3x + 3.

map_coefficients!(f::Function, p::AbstractPolynomialLike, nonzero = false)

Mutate p by replacing each coefficient α by f(α). The function may return zero in which case the term is dropped. If the function is known to never return zero for a nonzero input, nonzero can be set to true to get a small speedup. The function returns p, which is identically equal to the second argument.

See also map_coefficients and map_coefficients_to!.

Examples

Let p = 2x*y + 3x + 1, after map_coefficients!(α -> mod(3α, 6), p), p is equal to 3x + 3.

map_coefficients_to!(output::AbstractPolynomialLike, f::Function, p::AbstractPolynomialLike, nonzero = false)

Mutate output by replacing each coefficient α of p by f(α). The function may return zero in which case the term is dropped. If the function is known to never returns zero for a nonzero input, nonzero can be set to true to get a small speedup. The function returns output, which is identically equal to the first argument.

See also map_coefficients! and map_coefficients.

conj(x::AbstractPolynomialLike)

Return the complex conjugate of x by applying conjugation to all coefficients and variables.

real(x::AbstractPolynomialLike)

Return the real part of x by applying real to all coefficients and variables; for this purpose, every complex-valued variable is decomposed into its real- and imaginary parts.

See also imag.

imag(x::AbstractPolynomialLike)

Return the imaginary part of x by applying imag to all coefficients and variables; for this purpose, every complex-valued variable is decomposed into its real- and imaginary parts.

See also real.

isreal(p::AbstractPolynomialLike)

Returns true if and only if no single variable in p was declared as a complex variable (in the sense that isreal applied on them would be true) and no coefficient is complex-valued.

mindegree_complex(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Return the minimal total complex degree of the monomials of p, i.e., minimum(degree_complex, terms(p)).

mindegree_complex(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}}, v::AbstractVariable)

Return the minimal complex degree of the monomials of p in the variable v, i.e., minimum(degree_complex.(terms(p), v)).

minhalfdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Return the minmal half degree of the monomials of p, i.e., minimum(halfdegree, terms(p))

maxdegree_complex(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Return the maximal total complex degree of the monomials of p, i.e., maximum(degree_complex, terms(p)).

maxdegree_complex(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}}, v::AbstractVariable)

Return the maximal complex degree of the monomials of p in the variable v, i.e., maximum(degree_complex.(terms(p), v)).

maxhalfdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Return the maximal half degree of the monomials of p, i.e., maximum(halfdegree, terms(p))

extdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Returns the extremal total complex degrees of the monomials of p, i.e., (mindegree_complex(p), maxdegree_complex(p)).

extdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}}, v::AbstractVariable)

Returns the extremal complex degrees of the monomials of p in the variable v, i.e., (mindegree_complex(p, v), maxdegree_complex(p, v)).

exthalfdegree(p::Union{AbstractPolynomialLike, AbstractVector{<:AbstractTermLike}})

Return the extremal half degree of the monomials of p, i.e., (minhalfdegree(p), maxhalfdegree(p))

monomial_vector(X::AbstractVector{MT}) where {MT<:AbstractMonomialLike}

Returns the vector of monomials X in increasing order and without any duplicates.

Examples

Calling monomial_vector on $[xy, x, xy, x^2y, x]$ should return $[x^2y, xy, x]$.

monomial_vector(a, X::AbstractVector{MT}) where {MT<:AbstractMonomialLike}

Returns b, Y where Y is the vector of monomials of X in increasing order and without any duplicates and b is the vector of corresponding coefficients in a, where coefficients of duplicate entries are summed together.

Examples

Calling monomial_vector on $[2, 1, 4, 3, -1], [xy, x, xy, x^2y, x]$ should return $[3, 6, 0], [x^2y, xy, x]$.

monomial_vector_type(X::AbstractVector{MT}) where {MT<:AbstractMonomialLike}

Returns the return type of monomial_vector.

empty_monomial_vector(p::AbstractPolynomialLike)

Returns an empty collection of the type of monomials(p).

sort_monomial_vector(X::AbstractVector{MT}) where {MT<:AbstractMonomialLike}

Returns σ, the orders in which one must take the monomials in X to make them sorted and without any duplicate and the sorted vector of monomials, i.e. it returns (σ, X[σ]).

Examples

Calling sort_monomial_vector on $[xy, x, xy, x^2y, x]$ should return $([4, 1, 2], [x^2y, xy, x])$.

merge_monomial_vectors{MT<:AbstractMonomialLike, MVT<:AbstractVector{MT}}(X::AbstractVector{MVT}}

Returns the vector of monomials in the entries of X in increasing order and without any duplicates, i.e. monomial_vector(vcat(X...))

Examples

Calling merge_monomial_vectors on $[[xy, x, xy], [x^2y, x]]$ should return $[x^2y, xy, x]$.

conj(x::AbstractVector{<:AbstractMonomial})

Return the complex conjugate of x by applying conjugation to monomials.

real(x::AbstractVector{<:AbstractMonomial})

Return the real part of x by applying real to all monomials; for this purpose, every complex-valued variable is decomposed into its real- and imaginary parts. Note that the result will no longer be a monomial vector.

See also imag.

imag(x::AbstractVector{<:AbstractMonomial})

Return the imaginary part of x by applying imag to all monomials; for this purpose, every complex-valued variable is decomposed into its real- and imaginary parts. Note that the result will no longer be a monomial vector.

See also real.

isreal(p::AbstractVector{<:AbstractMonomial})

Returns true if and only if every single monomial in p would is real-valued.