Polynomrest

Polynomrest is a term used in algebra and computer algebra systems to denote the remainder component in polynomial arithmetic, and in some sources the name of a software package for modular polynomial computations.

Definition and concept: In a polynomial ring F[x] over a field F, fix a nonzero modulus m(x). The Polynomrest of a polynomial p(x) with respect to m(x) is the unique remainder r(x) with degree less than deg m such that p(x) ≡ r(x) mod m(x). The set of all such remainders forms the quotient ring F[x]/(m).

Structure and properties: The Polynomrest operator Rest_m acts as a projection from F[x] onto the set R of representatives of degree < deg m. With addition and multiplication performed modulo m, R becomes a finite-dimensional algebra isomorphic to the quotient ring F[x]/(m). If m is monic, Rest_m is well-defined and unique. The construction extends to multivariate polynomials and to modules over F[x] with respect to a submodule, yielding generalized reduction concepts.

Computational aspects: Remainder computation uses polynomial division, and many computer algebra systems implement modular reduction and arithmetic in quotient rings. For multivariate or large-scale problems, Gröbner bases provide a framework for reduction in more complex settings.

Applications: Polynomrest-based arithmetic underpins coding theory and digital communications, where arithmetic in quotient rings supports error-detecting and error-correcting codes, as well as certain cryptographic constructions. It also informs factorization and root-finding algorithms that operate modulo a fixed polynomial.

Variants and history: Different modulus choices yield different quotient rings; non-monic or non-field coefficient rings require additional care. The remainder concept is classical in polynomial algebra, with Polynomrest appearing in modern expositions and software documentation to emphasize reduction modulo a fixed polynomial.