  For real planar polynomial differential systems there appeared a simple
version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there
an upper bound on the number of algebraic limit cycles of all polynomial vector
fields of degree $m$?} In [J. Differential Equations, 248(2010), 1401--1409]
Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper
bound, in the case of invariant algebraic curves generic for the vector fields,
and they posed the following conjecture: {\it Is $1+(m-1)(m-2)/2$ the maximal
number of algebraic limit cycles that a polynomial vector field of degree $m$
can have?}
  In this paper we will prove this conjecture for planar polynomial vector
fields having only nodal invariant algebraic curves. This result includes the
Llibre {\it et al}\,'s as a special one. For the polynomial vector fields
having only non--dicritical invariant algebraic curves we answer the simple
version of the 16th Hilbert problem.
