Conditions for Additional Roots from Maximal-Rank Minors of Macaulay Matrices

Necessary conditions, under which the maximal-rank minors of a (possibly singular) Macaulay matrix of a polynomial system vanish, are analyzed. It is shown that the vanishing of the maximal-rank minors of the Macaulay matrix of a system of parametric polynomials under specialization is a necessary condition for the specialized polynomials to have an additional common root even when the parametric system has common roots without any specialization of parameters. For such a parametric system, its resultant is identically zero. A theorem of independent interest also gives a degree bound from which the Hilbert function of a certain zero-dimensional polynomial system that is not necessarily a complete intersection, as defined by Macaulay in his 1913 paper, becomes constant. These results are not only of theoretical interest, but it extends the class of parametric polynomial systems whose zeros can be analyzed using matrix based resultant formulations. Particularly, the main result has applications in areas where conditions for additional common roots of polynomial systems with generic roots are needed, such as in implicitization of surfaces with base points and in reasoning about geometric objects.