##  [Dehomogenization](/dehomogenization-1) 

 Definition

The algebraic operation that recovers an affine polynomial or system from a homogeneous (projective) polynomial by specializing the homogenizing variable to a nonzero constant (commonly 1) or by choosing an affine patch; it converts homogeneous equations in projective coordinates into ordinary polynomial equations in affine coordinates.

 

 

 

 

 

 





## Principle

Principle

Choose a projective coordinate (the homogenizing variable) that is invertible on a chosen affine chart and substitute a fixed nonzero value for it, thereby projecting the projective variety to an affine variety and eliminating the extra scaling degree of freedom.

 

 

 

 

 





## Demonstration

Demonstration

Given a homogeneous polynomial F(x0,x1,...,xn) of degree d, set x0 = 1 to obtain f(x1,...,xn) = F(1,x1,...,xn). For a system defining a projective curve, dehomogenizing with respect to x0 yields an affine system whose finite solutions correspond to projective solutions with x0 ≠ 0; solutions with x0 = 0 represent points at infinity and are not recovered by this patch. In numerical root finding one often dehomogenizes a homogenized system before applying affine solvers, noting that multiplicities at infinity may be lost or require coordinate changes to reveal.

 

 

 

 

## Misapplication

Misapplication

Blindly substituting 1 for the homogenizing variable without checking which projective points have that coordinate nonzero can hide solutions at infinity; using x0 = 0 as the substitution (or otherwise forcing a noninvertible value) produces nonsensical affine equations and can destroy multiplicity information or introduce spurious degeneracies.

 

 

 

 

 





## Consequence

Consequence

When applied correctly, dehomogenization yields an affine system amenable to standard polynomial manipulation and numerical methods, reduces the dimension of the ambient coordinate ring by one, and gives one chart of the projective variety; however it may omit points lying in other projective charts and can change the apparent multiplicity structure unless handled with care.

 

 

 

 

## Reversal

Reversal

Homogenization: introduce an extra homogenizing variable to convert affine polynomials into homogeneous ones so that projective techniques and degree-based constructions (resultants, homogeneous Macaulay matrices) become available; homogenization and dehomogenization are inverse only up to choice of affine chart and scaling.

 

 

 

 

 





## Boundary

Boundary

Applies only where a projective coordinate can be inverted (fields or rings where the chosen constant is a unit); does not capture projective points in the coordinate hyperplane where the chosen variable vanishes and does not universally preserve multiplicity data without additional local analysis or change of chart.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between the local affine viewpoint obtained by dehomogenization and the global, coordinate-free projective perspective: dehomogenization simplifies computation but sacrifices the uniform treatment of directions at infinity; choices of patch create noncanonical representations.

 

 

 

 

 





## Synthesis

Synthesis

Dehomogenization is the controlled specialization of the homogenizing coordinate to produce an affine representative of a projective system: it trades global projective invariance for a concrete affine model useful for computation, with the trade-offs being loss of information at the chosen hyperplane at infinity and sensitivity of multiplicity data to the patch choice.