Nonlinear algebra is the nonlinear analogue to

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

, generalizing notions of spaces and transformations coming from the linear setting. Algebraic geometry is one of the main areas of mathematical research supporting nonlinear algebra, while major components coming from

computational mathematics Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006 ...

support the development of the area into maturity. The topological setting for nonlinear algebra is typically the

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...

, where closed sets are the algebraic sets. Related areas in mathematics are

tropical geometry In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So fo ...

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

, and

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

Algebraic geometry

Nonlinear algebra is closely related to algebraic geometry, where the main objects of study include

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation ...

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...

, and schemes.

Computational nonlinear algebra

Current methods in computational nonlinear algebra can be broadly broken into two domains: symbolic and numerical. Symbolic methods often rely on the computation of Gröbner bases and

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over ...

s. On the other hand,

numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

typically use algebraically founded

homotopy continuation Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations. Homotopy con ...

, with a base field of the complex numbers.

References

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Algebraic geometry

Computational nonlinear algebra

References

See also