PolynomialSqrt

Status: Stable

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

Description

PolynomialSqrt[p]

gives a polynomial s with s^2 == p when p is a perfect square (every non-constant irreducible factor has even multiplicity; the numeric content is carried through Sqrt), and $Failed otherwise. PolynomialSqrt[p, x] treats p as a polynomial in x.

Examples (3)

Every input below was run against the current Mathilda build and its output recorded.

Applications (3)

(1 + x)^2

In[1]:= PolynomialSqrt[x^2 + 2 x + 1]
Out[1]= 1 + x

Returned in factored form

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

Not a perfect square

In[3]:= PolynomialSqrt[x^2 + 1]
Out[3]= $Failed

Implementation notes

Algorithm. builtin_polynomialsqrt returns a polynomial s with s^2 == p when p is a perfect square, and $Failed otherwise. The input is PolynomialSqrt[p] or PolynomialSqrt[p, x] (treating p as a polynomial in x). After Expand, a zero input maps to 0. Otherwise:

1. Factor p into constant * prod(base_i^e_i).
2. Walk the factors. A numeric or (in the 2-arg form) x-free factor is content and is multiplied into a running constant konst. A non-constant base needs even multiplicity e_i: take base_i^{e_i/2}. A non-constant factor of odd multiplicity (including a bare multiplicity-1 irreducible) means p is not a perfect square, so the whole thing fails.
3. Assemble s = Sqrt[konst] * prod(base_i^{e_i/2}) and emit it only after an exact certificate: Expand[s^2 - p] must be the zero polynomial. If the check fails (or any factor was odd-multiplicity), return $Failed.

The 2-arg form carries a symbolic/algebraic leading coefficient through Sqrt (e.g. Sqrt[-c] for -c x^2) rather than rejecting it as a bare irreducible — needed by Cherry's Erf-argument construction over an algebraically closed constant field.

Data structures. Expr trees built with the module's internal_factor and the expr_expand / eval_and_free helpers; the half-power factors are collected in a small Expr** array before being multiplied together.

Complexity / limits. Dominated by Factor[p]. Attributes: Protected. The square root is returned in factored form (e.g. (-1 + x)(1 + x), not the expanded x^2 - 1), and the numeric content is carried as an exact Sqrt when it is not a perfect square itself. Anything whose factorisation has an odd-multiplicity non-constant factor is $Failed.

Attributes: Protected.

References

• Source: src/poly/facpoly.c
• Specification index: Mathilda_spec.md
• Tests: tests/test_polynomialsqrt.c

Notes & additional examples

Notes

PolynomialSqrt[p] returns a polynomial s with s^2 == p when p is a perfect square — every non-constant irreducible factor must have even multiplicity, and the numeric content is carried through Sqrt — and $Failed otherwise. PolynomialSqrt[p, x] treats p as a polynomial in x, so any factor free of x counts as constant content.

The result is given in factored form: PolynomialSqrt[(x^2 - 1)^2] is (-1 + x)(1 + x), not the expanded x^2 - 1. Every success carries an exact certificate — Expand[s^2 - p] must be zero before s is returned — so a near-miss is reported as $Failed rather than an approximate root. PolynomialSqrt is Protected.