A Formal Study of Algebraic Constraint

We present a model for computation of algebraic constraint. An algebraic constraint is defined to be a boolean formula of equations in which every equation is expressed by a polynomial over a field, and hence such constraint may contain negation of an equation, that is, a form f 6= 0 where f is some polynomial. Algebraic constraint appears in many application fields of data analysis such as study of geometries, computer aided design, robotics and mechanics. It is wellknown that we can describe negations in a form of equations by using slack variables, but traditional approaches assume the same number of slack variables as that of negations. This means the dimension of the ambient space, in which the targeted manifold is embedded, depends on the number of negations used in the constraint, hence the dimension is not intrinsic in the constraint. We construct an algebraic model that enables to describe an arbitrary boolean formula including multiple negations by using only one slack variable. Also the model provides boolean operations that commute with algebraic operations of polynomials in a natural way, in which we introduce a kind of semiring and its operations in order to make do with one slack variable. To use one slack variable means that we can always consider constraints on the ambient space of the same dimension (n + 1) where n is the number of original variables. We present our approach to construct this model and show some important properties of it.