(Too long for a comment) Replacing $\xi$ with a variable $x$, Mathematica gives the following:

\begin{align*} \Delta(2)&=-1+x\\ \Delta(3)&=(-1+x)^3 x (1+x)\\ \Delta(4)&=(-1+x)^6 x^4 (1+x)^2 \left(1+x+x^2\right)\\ \Delta(5)&=(-1+x)^{10} x^{10} (1+x)^4 \left(1+x^2\right) \left(1+x+x^2\right)^2\\ \Delta(6)&=(-1+x)^{15} x^{20} (1+x)^6 \left(1+x^2\right)^2 \left(1+x+x^2\right)^3 \left(1+x+x^2+x^3+x^4\right) \end{align*} etc...

There is a nice pattern: for $d\ge3$, $\Delta(d)$ contains as factors exactly the cyclotomic polynomials of degree 1 to $d-1$, and $x$, raised to some powers.

Edit: I checked up to $d=12$ and, as Benjamin remarks, the exponents seems to be:

• triangular numbers for the factor $x-1$ (see https://en.wikipedia.org/wiki/Triangular_number)
• tetrahedral numbers for the factor $x$ (see https://en.wikipedia.org/wiki/Tetrahedral_number)

I searched in the OEIS for other exponent sequences, but it returns a lot of possibilities. For example, the exponents over $1+x$ go like this: 1,2,4,6,9,12,16,20,25,30,..., and the OEIS gives 16 results: https://oeis.org/search?q=1%2c2%2c4%2c6%2c9%2c12%2c16%2c20%2c25%2c30

(Too long for a comment) Replacing $\xi$ with a variable $x$, Mathematica gives the following:

\begin{align*} \Delta(2)&=-1+x\\ \Delta(3)&=(-1+x)^3 x (1+x)\\ \Delta(4)&=(-1+x)^6 x^4 (1+x)^2 \left(1+x+x^2\right)\\ \Delta(5)&=(-1+x)^{10} x^{10} (1+x)^4 \left(1+x^2\right) \left(1+x+x^2\right)^2\\ \Delta(6)&=(-1+x)^{15} x^{20} (1+x)^6 \left(1+x^2\right)^2 \left(1+x+x^2\right)^3 \left(1+x+x^2+x^3+x^4\right) \end{align*} etc...

There is a nice pattern: for $d\ge3$, $\Delta(d)$ contains as factors exactly the cyclotomic polynomials of degree 1 to $d-1$, and $x$, raised to some powers.

Edit: I checked up to $d=12$ and, as Benjamin remarks, the exponents seems to be:

• triangular numbers for the factor $x-1$ (see https://en.wikipedia.org/wiki/Triangular_number)
• tetrahedral numbers for the factor $x$ (see https://en.wikipedia.org/wiki/Tetrahedral_number)

I searched in the OEIS for other exponent sequences, but it returns a lot of possibilities. For example, the exponents over $1+x$ go like this: 1,2,4,6,9,12,16,20,25,30,..., and the OEIS gives 16 results: https://oeis.org/search?q=1%2c2%2c4%2c6%2c9%2c12%2c16%2c20%2c25%2c30

Another edit: Problem solved, see at the end of T. Amdeberhan's answer.

(Too long for a comment) Replacing $\xi$ with a variable $x$, Mathematica gives the following:

\begin{align*} \Delta(2)&=-1+x\\ \Delta(3)&=(-1+x)^3 x (1+x)\\ \Delta(4)&=(-1+x)^6 x^4 (1+x)^2 \left(1+x+x^2\right)\\ \Delta(5)&=(-1+x)^{10} x^{10} (1+x)^4 \left(1+x^2\right) \left(1+x+x^2\right)^2\\ \Delta(6)&=(-1+x)^{15} x^{20} (1+x)^6 \left(1+x^2\right)^2 \left(1+x+x^2\right)^3 \left(1+x+x^2+x^3+x^4\right) \end{align*} etc...

There is a nice pattern: for $d\ge3$, $\Delta(d)$ contains as factors exactly the cyclotomic polynomials of degree 1 to $d-1$, and $x$, raised to some powers.

(Too long for a comment) Replacing $\xi$ with a variable $x$, Mathematica gives the following:

\begin{align*} \Delta(2)&=-1+x\\ \Delta(3)&=(-1+x)^3 x (1+x)\\ \Delta(4)&=(-1+x)^6 x^4 (1+x)^2 \left(1+x+x^2\right)\\ \Delta(5)&=(-1+x)^{10} x^{10} (1+x)^4 \left(1+x^2\right) \left(1+x+x^2\right)^2\\ \Delta(6)&=(-1+x)^{15} x^{20} (1+x)^6 \left(1+x^2\right)^2 \left(1+x+x^2\right)^3 \left(1+x+x^2+x^3+x^4\right) \end{align*} etc...

There is a nice pattern: for $d\ge3$, $\Delta(d)$ contains as factors exactly the cyclotomic polynomials of degree 1 to $d-1$, and $x$, raised to some powers.

Edit: I checked up to $d=12$ and, as Benjamin remarks, the exponents seems to be:

• triangular numbers for the factor $x-1$ (see https://en.wikipedia.org/wiki/Triangular_number)
• tetrahedral numbers for the factor $x$ (see https://en.wikipedia.org/wiki/Tetrahedral_number)

I searched in the OEIS for other exponent sequences, but it returns a lot of possibilities. For example, the exponents over $1+x$ go like this: 1,2,4,6,9,12,16,20,25,30,..., and the OEIS gives 16 results: https://oeis.org/search?q=1%2c2%2c4%2c6%2c9%2c12%2c16%2c20%2c25%2c30