The operation of setting the homogenizing variable to a nonzero scalar (commonly 1) or otherwise restricting to an affine chart to recover an affine polynomial or affine variety from a homogeneous polynomial.

Principle

Dehomogenization evaluates the homogeneous polynomial on a chosen affine patch (e.g., z = 1) to pass from projective coordinates to affine coordinates; it is the local inverse of homogenization on that chart but discards information about the hyperplane where the homogenizing coordinate vanishes.

Demonstration

From the homogenized f_h(x,y,z) = x^2 + y z + z^2, dehomogenizing at z = 1 gives f(x,y) = x^2 + y + 1. Points with z = 0 correspond to points at infinity lost under this dehomogenization.

Misapplication

Plugging in the homogenizing variable as zero when intending to recover affine data, thereby losing the intended chart and misrepresenting points at infinity; or treating dehomogenization as a globally invertible operation rather than a local chart restriction.

Consequence

Dehomogenization produces affine equations suitable for local computation, solving and local analysis, but it omits the projective points on the hyperplane z = 0 and so cannot recover global projective data without additional patches.

Reversal

Homogenization reintroduces the homogenizing variable and restores projective-degree structure, producing a homogeneous polynomial whose restriction to the chosen chart recovers the affine polynomial.

Boundary

Requires the choice of a nonzero value for the homogenizing variable and consistent degree conventions; valid as an inverse only on the corresponding affine chart (the complement of the hyperplane where the homogenizing coordinate vanishes). Not applicable to polynomials that were not originally derived by homogenization unless degrees are adjusted.

Semantic Tension

Tension between dehomogenization as evaluation/substitution and as passage to an affine chart: evaluation at a scalar is an algebraic operation, while passing to a chart is geometric and local; also between different chart choices (z = 1 versus other nonzero values) which introduce scalings.

Synthesis

Dehomogenization is the local procedure that recovers affine representatives from homogeneous polynomials by fixing the homogenizing coordinate, enabling concrete computation on an affine chart while necessarily discarding points that lie on the hyperplane at infinity.