A tangent cone of a complex analytic set germ 
at the origin records its leading
homogeneous equations. If 
is the ideal of holomorphic
function germs vanishing on 
, form the ideal generated by the
lowest-degree nonzero homogeneous parts of all elements
of 
. Its zero
set is the tangent cone 
(Chirka 1989).
For a hypersurface 
with 
nonzero and vanishing
at the origin, this gives
 |
where 
is the first nonzero homogeneous part of the power
series of 
.
For a general analytic germ, the initial parts must be taken
from all elements of the defining ideal,
not just an arbitrary list of generators. Tangent cones are useful in the Zariski
multiplicity conjecture and its metric version (Sampaio 2026).
