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[3] [4] In the twentieth century, polynomial regression played an important role in the development of regression analysis, with a greater emphasis on issues of design and inference. [5] More recently, the use of polynomial models has been complemented by other methods, with non-polynomial models having advantages for some classes of problems.
In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
The graph of a polynomial function of degree 3. The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate").
Polynomial models have well known and understood properties. Polynomial models have moderate flexibility of shapes. Polynomial models are a closed family. Changes of location and scale in the raw data result in a polynomial model being mapped to a polynomial model. That is, polynomial models are not dependent on the underlying metric.
The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.
The Barth surface, shown in the figure is the geometric representation of the solutions of a polynomial system reduced to a single equation of degree 6 in 3 variables. Some of its numerous singular points are visible on the image. They are the solutions of a system of 4 equations of degree 5 in 3 variables.
Elimination theory culminated with the work of Leopold Kronecker, and finally Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which are described in the chapter on Elimination theory in the first editions (1930) of van der Waerden's Moderne Algebra.