laurent#

Utilities for manipulating (Laurent) polynomials with SymPy: - reciprocal transforms f(x) -> f(x^{-1}) - normalization for ordinary and Laurent polynomials - symmetric normalization under t ↔ 1/t - canonicalization under permutations of variables

Functions

`canonicalize_under_variable_permutation`(expr)	Canonicalize an expression up to permutations of variables (and optionally sign).
`extract_variables`(expr[, prefix])	Extract variables of the form <prefix><digits> from a SymPy expression.
`normalize_laurent_polynomial`(expr[, ...])	Normalize a (Laurent) polynomial up to monomials and ±1.
`normalize_polynomial`(poly)	Normalize a multivariate (non-Laurent) polynomial by removing minimal exponents.
`normalize_symmetric`(expr, variable)	Normalize a Laurent polynomial symmetrically in a single variable.
`reciprocal`(expr, var)	Return the reciprocal transform of a polynomial in a given variable.

reciprocal(expr, var)#

Return the reciprocal transform of a polynomial in a given variable.

Applies the substitution var → var**(-1) and expands.

Parameters:

expr (Expr) – SymPy expression.
var (Symbol | str) – The variable to invert (Symbol or its name).

Returns:

Expr – The transformed expression.

Return type:

Expr

normalize_polynomial(poly)#

Normalize a multivariate (non-Laurent) polynomial by removing minimal exponents.

For each generator, divides the polynomial by the minimal exponent of that generator across all monomials, so exponents start at 0.

Parameters:: poly (Poly) – SymPy Poly with nonnegative exponents.
Returns:: Poly – A normalized polynomial with minimal support.
Return type:: Poly

normalize_laurent_polynomial(expr, variables=None, normalize_sign=True)#

Normalize a (Laurent) polynomial up to monomials and ±1.

Steps:

Shift exponents of selected variables so that all are nonnegative.
Optionally flip overall sign so the leading term has positive coefficient.

Parameters:

expr (Expr) – SymPy expression (may have negative exponents).
variables (Iterable[Symbol] | None) – Variables to consider. If None, use all free symbols.
normalize_sign (bool) – If True, ensure leading term’s coefficient is positive.

Returns:

Expr – A canonicalized expression.

Return type:

Expr

normalize_symmetric(expr, variable)#

Normalize a Laurent polynomial symmetrically in a single variable.

The result is centered and made symmetric under variable ↔ 1/variable, then the overall sign is chosen so the leading term is positive.

Parameters:

expr (Expr) – SymPy expression.
variable (Symbol) – The variable to symmetrize.

Returns:

Expr – Symmetrically normalized expression.

Return type:

Expr

extract_variables(expr, prefix=None)#

Extract variables of the form <prefix><digits> from a SymPy expression.

Examples

prefix=None → returns all symbols like t1, x2, a10, grouped by (prefix, index)
prefix=”t” → returns only t1, t2, …

Parameters:

expr (Expr) – SymPy expression.
prefix (str | None) – Optional prefix filter.

Returns:

list[Symbol] – Symbols sorted by (prefix, numeric index).

Return type:

list[Symbol]

canonicalize_under_variable_permutation(expr, variables=None, allow_sign_change=False)#

Canonicalize an expression up to permutations of variables (and optionally sign).

For all permutations of the selected variables, compute the permuted polynomial and choose the lexicographically minimal representation of (exponent-vector → coefficient) pairs (optionally also considering the negated polynomial).

Parameters:

expr (Expr) – SymPy expression.
variables (Sequence[Symbol] | None) – Sequence of variables to permute. If None, attempts to detect with extract_variables() and then sorts them.
allow_sign_change (bool) – If True, also consider -f under permutations and pick the best.

Returns:

Expr – Canonical representative under variable permutations (and ±1 if enabled).

Return type:

Expr