The applications of resultants and the Bernshtein formula for the dimensional synthesis of linkage components for finite precision positions are discussed. The closed-form solutions, which are derived from systems of polynomials in multiple unknowns by applying resultant theory, are in forms of polynomial equations of a single unknown. For the case of two compatibility equations, the closed form solution is a sixth degree solution polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree solution polynomial. For each case, the Bernshtein formula is applied to calculate the number of solutions of the system of polynomial equations. The calculated numbers of solutions match the degrees of the solution polynomials for both cases.

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