# On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $$R=\mathbb C[x,y]$$.

Consider the matrix $$A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&x^5+y^5&5x^5&10x^5\\10y^5&10y^5&5y^5&x^5+y^5&5x^5\\5y^5&10y^5&10y^5&5y^5&x^5+y^5 \end{pmatrix} \in M_5(R)$$.

Also consider the polynomial $$p(x,y)=\det (A-I)\in R=\mathbb C[x,y]$$.

How to show that $$p(x,y)$$ can be factored into $$5^2=25$$ linear polynomials in $$x$$ and $$y$$ (over $$\mathbb C$$) ?

If $$x+y=1$$, then it can be shown that $$1$$ is an an eigenvalue of $$A$$, so the image of $$p(x,y)$$ in $$\mathbb C[x,y]/(x+y-1)$$ is $$0$$; thus $$x+y-1$$ is a factor of $$p(x,y)$$; but other than that, I'm unable to say anything else.

I feel I some how have to apply Hilbert Nullstellensatz, but I don't know how.

Define the $$\, n\times n\,$$ matrix $$\, A = \{a_{i, j}\}_{i, j=1}^n \,$$ where $$\, a_{i, j} = \binom{n}{j-i}x^n + \binom{n}{i-j}y^n. \,$$ The matrix $$A$$ is a special Toeplitz matrix. Let $$\, p(x, y) := \det(A-I).\,$$ Since $$\, x + y - 1 \,$$ is a factor, then also $$\, z x + w y - 1 \,$$ is a factor where $$\, z, w \,$$ are any pair of $$\,n$$th roots of unity and there are $$\,n^2\,$$ pairs.
• Right. The matrix stays the same when $x,y$ are replaced by $zx,wy$. Sep 24, 2018 at 6:15
• Took a few tries, I did $n=3$ by combining into real products, once I fixed some errors it was nice , had pari check the product against the determinant // f3 = x^3 + y^3 + 3 * x * y - 1 // f5 = x^2 + 2 * x * y + y^2 + x + y + 1 // f7 = x^2 - x * y + y^2 + x - 2 * y + 1 // f9 = x^2 - x * y + y^2 -2 * x + y + 1 // p = x^9 + (3*y^3 - 3)*x^6 + (3*y^6 + 21*y^3 + 3)*x^3 + (y^9 - 3*y^6 + 3*y^3 - 1) // Sep 24, 2018 at 18:15
• Using $1 + w + w^2 = 0,$ the term I called f3 is $$(x+y-1)(xw+yw^2-1)(x w^2 + y w - 1)$$ the other terms f5,f7,f9 are products of pairs of complex conjugate terms (assuming x,y real for cosmetic purposes) Sep 24, 2018 at 18:22