Polynomsystem

Polynomsystem refers to a collection of polynomial equations in one or more variables. Formally, a polynomsystem over a field F consists of polynomials p1(x1,...,xn)=0, ..., pm(x1,...,xn)=0 and the problem is to find common solutions (a1,...,an) in an algebraic extension of F that satisfy all equations.

The set of solutions is the affine variety V(I) corresponding to the ideal I generated by the polynomials. The study of polynomsystems is central to algebraic geometry, linking algebraic properties of the polynomials to geometric properties of their solution sets. Over algebraically closed fields, the correspondence is sharpened by the Nullstellensatz, which relates ideals to their zero sets.

There are linear and nonlinear cases. Linear polynomsystems are solvable by standard linear algebra. In general, polynomial systems are nonlinear and may have zero, finitely many, or infinitely many solutions; they can be discrete or form higher-dimensional varieties. The structure of the solution set depends on the number of equations, the number of variables, and the degrees of the polynomials.

Solving methods: symbolic approaches such as Gröbner bases (Buchberger’s algorithm) enable elimination of variables and decision of consistency; resultants and elimination theory play a similar role. Numerical approaches include homotopy continuation and Newton-type methods that approximate solutions, often used when exact symbolic solutions are impractical.

Complexity and theory: the general problem of solving polynomial systems is computationally hard; Gröbner basis computation can be doubly exponential in the worst case, and many decision problems are NP-hard or involve undecidable aspects in certain settings. The algebraic-geometric perspective links I to the geometry of V(I) via dimension, degree and multiplicity.

Applications: robotics and kinematics, computer-aided design and vision, cryptography and error-correcting codes, chemistry and physics, and many areas of optimization rely on solving polynomial systems. See also polynomial equation, algebraic geometry.