Polinomit

Polinomit is a term sometimes used in mathematical literature to denote a polynomial whose coefficients depend polynomially on one or more parameters. In standard terminology, such objects are called parametrized polynomials or polynomials with parametric coefficients; polinomit remains a rarely used synonym.

Let F be a field and θ = (θ1, ..., θm) parameters. A polinomit P is an element of F[θ][x], i.e., a polynomial in x with coefficients in F[θ]. For example, P(x, θ) = sum_{i=0}^d c_i(θ) x^i, where each c_i ∈ F[θ].

Specialization and operations: For fixed θ0, P(x, θ0) ∈ F[x] is the specialization. The set of all polinomits forms the polynomial ring F[θ][x]. Arithmetic is performed coefficient-wise, and properties such as factorization, roots, and discriminants can be studied after specialization or in a symbolic form.

Applications: Polinomits are used to model families of polynomials depending on parameters in elimination theory and algebraic geometry, to study how roots vary with parameters, and in computational algebra when solving polynomial systems with parameters. They provide a convenient notation for discussing parametric behavior without fixing the parameters prematurely.

Example: P(x, t) = x^2 + (t+1)x + t^2. As t varies, this yields a family of quadratics whose roots and discriminant depend polynomially on t, illustrating how a single polinomit encodes a whole parameterized family.

Notes: The term polinomit is not standard; in most literature the same object is called a parametrized polynomial. Related terms include polynomial with parameters, polynomial family, and elimination theory.