Squares of polynomials with all nonzero integer coefficients

Question

For polynomials of one variable with only and all nonzero integer coefficients, do there exist any where the polynomial consists of more than one term, and the square of the polynomial has the same number of terms or fewer than the polynomial?

For instance (to contrast), $(x + 1)^2 = x^2 + 2x + 1$

The polynomial in parentheses has two terms, but the square of the polynomial has three terms.

Another instance, $(x^2 - x + 2)^2 = x^4 - 2x^3 + 5x^2 - 4x + 4$

The polynomial in parentheses has three terms, but the square of the polynomial has five terms.

Note: This is not a duplicate of another question. That merely requested that that [p(x)]^2 had to have fewer nonzero terms than p(x). Mine requires that p(x) 1) must have all of its terms' coefficients be nonzero and integers, and 2) the number of terms could be equal, not just fewer.

Answer 1

Yes, and apparently the minimal examples occur in degree $12$. Coppersmith and Davenport found a class of examples: Define $$p_{\alpha}(x) := (125 x^6 + 50 x^5 - 10 x^4 + 4 x^3 - 2 x^2 + 2 x + 1) (\alpha x^6 + 1) .$$ Expanding shows that none of the $13$ coefficients are $0$ provided $\alpha \neq 0, -125$, but for eight particular values $\alpha$, including $\alpha = -110$, $p_\alpha(x)^2$ has precisely 12 nonzero coefficients.

Coppersmith, D. and Davenport, J. "Polynomials Whose Powers Are Sparse." Acta Arith. 58, 79-87, 1991.

Answer 2

Here is an example that answers my question in the case where [p(x)]^2 has the same number of terms as p(x):

p(x) = x^4 + 2x^3 - 2x^2 + 4x + 4

[p(x)]^2 = x^8 + 4x^7 + 28x^4 + 32x + 16