Dehomogenization and the affine gradient criterion #
Setting the i-th coordinate to one turns a polynomial in n + 1 variables into a
polynomial in n variables. The remaining coordinates are indexed by i.succAbove.
We compute evaluation and partial derivatives of this substitution, and prove that at
a zero of a homogeneous equation its full gradient vanishes exactly when the gradient
of the affine equation vanishes. Thus the usual nonsingularity test can be checked in
any standard affine chart of a projective hypersurface; for a plane curve, take n = 2.
Euler's identity recovers the omitted partial derivative. No invertibility of the degree is needed, so the criterion also applies when the characteristic divides the degree.
Conversely, a homogeneous polynomial is recovered from its degree and its equation in one chart
(TauCeti.dehomogenize_injOn), and every polynomial of total degree at most N is the equation of
some homogeneous polynomial of degree N (TauCeti.exists_isHomogeneous_dehomogenize_eq).
Together they lift a factorization in an affine chart to a factorization of the homogeneous
equation, up to a power of the omitted coordinate.
Combined with the weak Nullstellensatz, the pointwise criterion gives an identity of ideals:
if every point of a projective hypersurface over an algebraically closed extension K of k
lying in a standard affine chart is nonsingular, then in that chart the equation and its partial
derivatives generate the unit ideal of the polynomial ring over k
(TauCeti.span_insert_range_pderiv_dehomogenize_eq_top).
References #
- W. Fulton, Algebraic Curves, Chapter 5, Section 1.
The substitution setting coordinate i to one and keeping the other coordinates,
indexed in increasing order by i.succAbove.
Instances For
Variables are substituted by inserting one at the chosen coordinate.
Dehomogenizing a monomial drops the exponent of the chosen coordinate and keeps the others,
indexed by i.succAbove.
Substituting after adjoining the omitted variable recovers the original polynomial. In particular, every affine polynomial is the dehomogenization of a polynomial.
Evaluation in an affine chart at a point of an R-algebra is evaluation of the homogeneous
coordinates with the chosen coordinate equal to one.
Evaluation in an affine chart is evaluation of the homogeneous coordinates with the chosen coordinate equal to one.
Dehomogenization does not increase the total degree.
For a homogeneous p of degree d and an exponent s of total degree d, the coefficient
of p at s is the coefficient of the affine equation at the restriction of s to the retained
coordinates. The i-th entry of s is determined by the others, so no information is lost.
For a homogeneous p of degree d, the coefficient of X_j ^ d in the affine equation is
the coefficient of the corresponding power X_{i.succAbove j} ^ d in p.
Dehomogenization is injective on homogeneous polynomials of a fixed degree: a homogeneous polynomial is determined by its degree and its equation in any standard affine chart.
Homogenization: every polynomial p in n variables of total degree at most N is the
equation in the i-th standard affine chart of a homogeneous polynomial of degree N in n + 1
variables, obtained by padding each homogeneous component of p with a power of X_i.
Changing coefficients commutes with dehomogenization.
Partial differentiation in a retained coordinate commutes with dehomogenization.
Euler's identity at a zero in an affine chart: the omitted partial derivative plus the coordinate-weighted sum of the retained partial derivatives is zero.
At a zero of a homogeneous equation, the homogeneous gradient vanishes if and only if the gradient of the equation in any chosen affine chart vanishes.
A homogeneous gradient is nonzero at a zero of the equation exactly when the affine gradient is nonzero. Over a field, this is the nonsingularity criterion.
At a zero of a homogeneous equation, the omitted partial derivative is determined by the retained partial derivatives.
At a homogeneous point whose i-th coordinate is nonzero, evaluation in the
standard affine chart differs by the inverse d-th power of that coordinate.
Over a semifield, a nonzero homogeneous coordinate can be normalized to one without changing whether the gradient vanishes. The derivatives of the affine equation are evaluated at the coordinate ratios.
Over a field, nonsingularity of a point of a homogeneous hypersurface is equivalent to a nonzero partial derivative of the equation in any affine chart containing it. For a plane equation this applies uniformly to all three standard charts.
Nonsingularity of a projective hypersurface, chart by chart: let p be homogeneous over
k, and suppose that at every zero of p in K^(n + 1) with i-th coordinate 1, for an
algebraically closed extension K of k, some partial derivative of p is nonzero. Then in the
i-th standard affine chart the equation and its partial derivatives generate the unit ideal of
k[X₁, …, Xₙ].