Ærm2ÆṛJ}¹?×AṪ½Ɗ×þ`ŒdṙLƊṚ§⁼
A monadic Link that accepts a non-empty list of coefficients ordered by degree* and yields 1 if the input represents the square of an integer polynomial, or 0 if not.
* i.e. the reverse order to the examples in the question, and \$0\$ being [0] rather than []
Try it online!
How?
Same approach as loopy walt's Python answer:
- get the polynomial with every other root of the polynomial defined by the input
- rescale using the square root of the coefficient of the highest degree in the input
- get the coefficients of the square of that polynomial
- check if that is equal to the input
Ærm2ÆṛJ}¹?×AṪ½Ɗ×þ`ŒdṙLƊṚ§⁼ - Link: list of coefficients, C
Ær - get the roots of the polynomial P defined by C
m2 - modulo-2 slice
? - if...
¹ - ...condition: no-op (falsey if P is just a constant)
Æṛ - ...then: get polynomial coefficients
} - ...else: use the right argument (implicitly C) with:
J - range of length (in this case that's [1])
(call this list NewCoefficients)
Ɗ - last three links as a monad - f(C):
A - absolute values
Ṫ - tail (absolute of coefficient of highest degree)
½ - square root
× - (NewCoefficients) multiplied by (that)
` - use as both arguments of:
þ - table of:
× - multiplication
Ɗ - last three links as a monad - f(X=that)
Œd - anti-diagonals of X (starts with main, sweeps South-East)
(has the effect of grouping product coefficients by degree)
L - length of X
ṙ - rotate (the anti-diagonals) left by (length of X)
(ensures that the groups are sorted by degree descending)
Ṛ - reverse
§ - sums (sum each group of like-degree product coefficients)
⁼ - equals C?
