The Thom encoding of a real root 
of a polynomial 
of polynomial degree 
is the vector
of signs of its successive derivatives,
 |
Thom's lemma states that distinct real roots of 
have distinct Thom encodings. A Thom
encoding therefore provides an exact representation of a real
root without using root isolation (Coste and
Roy 1988).
The same idea can isolate branches of an algebraic function. Let 
be a polynomial of polynomial
degree 
in 
. On a connected
set where the resultants
 |
the signs of the 
-derivatives at each real
root are nonzero and constant. The equation 
together with the corresponding derivative-sign inequalities then selects
a unique continuous branch (Maaz and Strzeboński
2025).
