Other Helpful Functions

compare_rational_func_by(f, g, by)

Returns

• -1 if f < g,
• 0 if f = g, and
• 1 if f > g.

Functions' numerators and denominators are compared using by.

decompose_derivative(varname, prefixes)

Determines if it is possible to represent the varname as a_number where a is an element of prefixes If yes, returns a pair (a, number), otherwise nothing

dennums_to_fractions(dennums)

Returns the field generators represented by fractions.

Input: an array of arrays of polynomials, as in [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

Output: an array of fractions [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

eval_at_dict(f, d)

Evaluates a polynomial/rational function on a dictionary of type var => val and missing values are replaced with zeroes

extract_coefficients(poly, variables)

Input:

• poly - multivariate polynomial
• variables - a list of variables from the generators of the ring of poly

Output:

• dictionary with keys being tuples of length length(variables) and values being polynomials in the variables other than those which are the coefficients at the corresponding monomials (in a smaller polynomial ring)

fractions_to_dennums(fractions)

Returns the field generators represented by lists of denominators and numerators.

Input: an array of fractions, as in [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

Output: an array of arrays of polynomials, [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

gen_tag_name(base; stop_words)
gen_tag_names(n, base; stop_words)

Generates a string which will not collide with the words in stop_words.

Arguments

• n: Generates a sequence of unique strings of length n
• base: A string or a vector of strings, the base for the generated sequence
• stop_words: A vector of strings, stop words

jacobian(ratfuncs, point)

Computes the evaluation of the jacobian of ratfuncs at point point

make_substitution(f, var_sub, val_numer, val_denom)

Substitute a variable in a polynomial with an expression

Input:

• f - the polynomial
• var_sub - the variable to be substituted
• var_numer - numerator of the substitution expression
• var_denom - denominator of the substitution expression

Output:

• polynomial - result of substitution

parent_ring_change(poly, new_ring)

Converts a polynomial to a different polynomial ring Input

• poly - a polynomial to be converted
• new_ring - a polynomial ring such that every variable name appearing in poly appears among the generators

Output:

• a polynomial in new_ring “equal” to poly

replace_with_ic(ode::ODE, funcs)

Takes an ode and a list of functions in the states and parameters and makes a change of variable names x(t) -> x(0). Function is used to prepare the output for the case of known initial conditions

select_pivots(M::MatElem)

Takes as input a matrix M in the reduced row echelon form and returns tow lists: of the pivot indices and the non-pivot ones

switch_ring(v, ring)

For a variable v, returns a variable in ring with the same name

uncertain_factorization(f)

Input:

• f - polynomial with rational coefficients

Output:

• list of pairs (div, certainty) where
  • div's are divisors of f such that f is their product with certain powers
  • if certainty is true, div is $Q$-irreducible