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PolynomialQuotient

Status: Stable

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

Description

PolynomialQuotient[p, q, x] gives the quotient of p and q, treated as polynomials in x, with any remainder dropped.
Option: Extension -> alpha (default None) divides over Q(alpha) using the Q(alpha)[x] long-division substrate; Sqrt[c], c^(1/n), and I are recognised forms for alpha.
Extension -> None and Extension -> Automatic are accepted and currently behave as the default (no extension).

Examples

All examples below are verified against the current Mathilda build.

In[1]:= PolynomialQuotient[x^4+2x+1, x^2+1, x]
Out[1]= -1 + x^2

In[2]:= PolynomialQuotient[x^2+2x+1, x^3, x]
Out[2]= 0

In[3]:= PolynomialQuotient[x^2+x+1, 2x+1, x]
Out[3]= 1/4 + 1/2 x

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

In[5]:= PolynomialQuotient[x^3 - 2, x - 2^(1/3), x, Extension -> 2^(1/3)]
Out[5]= 2^(2/3) + 2^(1/3) x + x^2

In[6]:= PolynomialQuotient[x^2 + 1, x - I, x, Extension -> I]
Out[6]= I + x

Implementation notes

Algorithm. builtin_polynomialquotient strips an optional Extension -> α/Automatic (routing to polynomialdivrem_with_extension over Q(α) when applicable), then calls the shared poly_div_rem(p, q, x, &Q, &R) and returns the expanded quotient Q, discarding R. poly_div_rem is textbook field long division in x: it expands both operands, repeatedly forms the next quotient term (lc(R)/lc(q)) · x^(deg R − deg q), subtracts term·q from the running remainder R, and stops when deg R < deg q. A fast path detects exact integer/bigint leading-coefficient divisions (via mpz_tdiv_qr) so the subtraction step never needlessly introduces rationals; otherwise quotient coefficients are formed symbolically. Coefficients are extracted with get_coeff_expanded against the already-expanded divisor.

Data structures. Expr polynomial trees in expanded form; quotient and remainder are returned through out_Q/out_R pointers.

  • Protected.
  • Default path uses polynomial long division over the field of rational functions in the coefficients.
  • Option Extension -> alpha (default None) lifts $p, q$ into $\mathbb{Q}(\alpha)[x]$ and runs the Q($\alpha$)-aware long division (qaupoly_divrem). Recognised forms for $\alpha$: Sqrt[c], c^(1/n), and I. Extension -> None and Extension -> Automatic are accepted and currently behave as the default. The extension path requires univariate input (a single live polynomial variable other than the alpha generator); multivariate inputs fall through to the standard path.
  • Threading Extension here keeps the polynomial arithmetic in the Q($\alpha$)[x] substrate and avoids the multivariate Q[$\alpha$, x] subresultant-PRS path that is exponentially slow on Sqrt[$\alpha$]-laden coefficients.

Attributes: Protected.

Implementation status

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

References

Notes & additional examples

Worked examples

In[1]:= PolynomialQuotient[x^2 - 1, x - 1, x]
Out[1]= 1 + x

In[1]:= PolynomialQuotient[x^3 + 2 x^2 + x + 1, x + 1, x]
Out[1]= x + x^2

In[1]:= PolynomialQuotient[x^2 + 1, x, x]
Out[1]= x

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

Notes

PolynomialQuotient[p, q, x] performs long division of p by q in the variable x and returns only the quotient, discarding the remainder. So (x^2 + 1) / x yields x (dropping the 1 remainder) and x^2 - 1 divides exactly by x - 1 to give 1 + x. Pair it with PolynomialRemainder to recover the full relation p = q*quotient + remainder. The Extension -> alpha option divides over Q(alpha); the default None/Automatic divides over the rationals.