Can there exist a polynomial on $w,x,y,z$ whose value is zero on the edges of a tesseract – or rather, on the projection of those edges to the unit sphere – and nonzero everywhere else on the unit sphere?
The dual's vertex figure has central symmetry, so its edges form great circles, leading to a simple answer: $(w^2+x^2) (w^2+y^2) (w^2+z^2) (x^2+y^2) (x^2+z^2) (y^2+z^2)$. That's not true of the tesseract: the great circles that include its edges also include each cell's major diagonals.