PolynomialLCM¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

PolynomialLCM[poly1, poly2, ...] gives the least common multiple of the polynomials.
Option Extension -> alpha computes the LCM over Q(alpha) via
lcm(a, b) = a*b / PolynomialGCD[a, b, Extension -> alpha].
Default Extension -> None computes over the rationals.

Examples¶

All examples below are verified against the current Mathilda build.

In[1]:= PolynomialLCM[(1+x)^2(2+x)(4+x), (1+x)(2+x)(3+x)]
Out[1]= (2 + x) (3 + x) (4 + x) (1 + x)^2

In[2]:= PolynomialLCM[x^4-4, x^4+4 x^2+4]
Out[2]= (-2 + x^2) (4 + 4 x^2 + x^4)

In[3]:= PolynomialLCM[x - Sqrt[2], x + Sqrt[2], Extension -> Sqrt[2]]
Out[3]= -2 + x^2

Implementation notes¶

Algorithm. builtin_polynomiallcm mirrors PolynomialGCD: it strips an optional Extension -> α/Automatic (routing to polynomiallcm_with_extension / qa_polynomiallcm_with_tower* over Q(α)), force-rationalises and re-numericalises inexact coefficients, and otherwise pre-decomposes each input with decompose_to_bp to handle numeric content and shared non-numeric factors. The polynomial parts are combined pairwise via the identity lcm(a, b) = a·b / gcd(a, b), where the GCD comes from poly_gcd_internal (the recursive multivariate subresultant PRS — see PolynomialGCD) and the exact division is done by exact_poly_div/Cancel. Multiple arguments fold left-to-right, accumulating the running LCM. When the exact quotient by the GCD can't be certified the code falls back to the plain product a·b.

Data structures. Expr trees throughout; BPList for the base/power decomposition; QAUPoly/QATower on the algebraic-extension path.

Protected, Listable.
Handles univariate and multivariate polynomials.
Treats algebraic numbers (like I) as independent variables or constants seamlessly during complex arithmetic evaluations.
Preserves explicit factored forms where possible.
Option Extension -> alpha computes the LCM over Q(alpha) via lcm(a, b) = a*b / PolynomialGCD[a, b, Extension -> alpha], returning the monic, expanded form. Same scope and fallback as PolynomialGCD's extension option.

Attributes: Listable, Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

G. E. Collins, "Subresultants and Reduced Polynomial Remainder Sequences", JACM 14(1), 1967.
Source: src/poly/poly.c
Specification: docs/spec/builtins/structural-manipulation.md

Notes & additional examples¶

Worked examples¶

In[1]:= PolynomialLCM[x^2 - 1, x^2 + 2 x + 1]
Out[1]= (-1 + x) (1 + 2 x + x^2)

In[1]:= PolynomialLCM[x^6 - 1, x^4 - 1]
Out[1]= (-1 + x^6) (1 + x^2)

In[1]:= PolynomialLCM[x^2 - 2, x^2 - Sqrt[2], Extension -> Sqrt[2]]
Out[1]= 2 Sqrt[2] - 2 x^2 - Sqrt[2] x^2 + x^4

Notes¶

PolynomialLCM returns the least common multiple of its polynomial arguments, computed as a b / gcd(a, b). For x^2 - 1 = (x-1)(x+1) and (x+1)^2 the shared factor x+1 appears only once in the LCM, so the result factors as (x-1)(x+1)^2. For x^6-1 and x^4-1 the common part is x^2-1, leaving (x^6-1)(x^2+1). The Extension -> alpha option computes the LCM over Q(alpha); with alpha = Sqrt[2] the inputs x^2-2 and x^2-Sqrt[2] are coprime there, so the LCM is their full product (x^2-2)(x^2-Sqrt[2]).