The authors study \(C^\infty\) one-parameter families of area-preserving mappings of \(\mathbb{R}^2\), \(f : \mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2\). An elementary \(n\)-furcation occurs at a point \((x,\mu)\) such that \(x\) is a periodic point of least period \(p\) for \(f_\mu\), the eigenvalues of \(Df^p_\mu (x)\) are \(n\)-th roots of unity, and some genericity hypotheses hold [\textit{K. R. Meyer}, Trans. Am. Math. Soc. 149, 95-107 (1970; Zbl 0198.429)]. If we set \(q = np\), then \(Df^q_\mu = I\). The authors calculate the multiplicity \(M_n\) of an elementary \(n\)- furcation point as a solution of the system \[ f^q_\mu(x) - x = 0,\quad \text{det} (Df^q_\mu (x) - I) = 0.\tag{1} \] The left hand side of (1) is thought of as a mapping from \(\mathbb{C}^3\) to \(\mathbb{C}^3\) (if \(f\) is not analytic, the Taylor series of \(f\) is truncated at an appropriate level), and multiplicity is in the sense of algebraic geometry. The authors find that \[ M_1 = 1,\;M_2 = 3,\;M_3 = 8,\quad \text{and} \quad M_n = n^2 + 2 \quad \text{for} \quad n \geq 4. \] For the area-preserving Hénon family \[ f_\mu \left ( \begin{smallmatrix} x \\ y \end{smallmatrix} \right) = \left( \begin{smallmatrix} \mu - y - x^2 \\ x \end{smallmatrix} \right), \tag{2} \] the authors show that for each \(q \in \mathbb{N}\), the total multiplicity in \(\mathbb{C}^3\) of all solutions of (1) is \(q2^{q - 1}\). They conclude that the known (real) bifurcation diagram of (2) exhibits all complex periodic points of least period 1 through 4, but this is not true for period 5.