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Wallis's constant is the real solution (OEIS A007493) to the cubic equationIt was solved by Wallis to illustrate Newton's methodfor numerical equation solving.

A root having multiplicity is called a simple root. For example, has a simple root at , but has a root of multiplicity 2 at , which is therefore not a simple root.

Newton's method for finding roots of a complex polynomial entails iterating the function , which can be viewed as applying the Euler backward method with step size unity to the so-called Newtonian vector field . The rescaled and desingularized vector field then has sinks at roots of and has saddle points at roots of that are not also roots of . The union of the closures of the unstable manifolds of the saddles of defines a directed graph whose vertices are the roots of and of , and whose edges are the unstable curves oriented by the flow direction. This graph, along with the labelling of each vertex with the multiplicity of as a root of , is defined to be the Newtonian graph of (Smale 1985, Shub et al. 1988, Kozen and Stefánsson 1997).

Given two functions and analytic in with a simple loop homotopic to a point in , if for all on , then and have the same number of roots inside .A stronger version has been proved by Estermann (1962). The strong version also has a converse, as shown by Challener and Rubel (1982).

Calculating the 13th root of a large number (that is a perfect 13th power) is a famous mental calculation challenge. However, because of difficulties in standardizing the time taken to find the root, the Guinness Book of World Records no longer maintains an entry for the 13th root.The official record (as of August 2005) is 13.55 seconds for a 100-digit perfect 13th power, as calculated by Alexis Lemaire on May 10, 2002. (By comparison, the Wolfram Language computes such roots in about 51 microseconds.) The record for a 200-digit number was also set by Lemaire, at 267.77 seconds (whereas the Wolfram Language takes roughly 82 microseconds).

The roots (sometimes also called "zeros") of an equationare the values of for which the equation is satisfied.Roots which belong to certain sets are usually preceded by a modifier to indicate such, e.g., is called a rational root, is called a real root, and is called a complex root.The fundamental theorem of algebra states that every polynomial equation of degree has exactly complex roots, where some roots may have a multiplicity greater than 1 (in which case they are said to be degenerate). In the Wolfram Language, the expression Root[p(x), k] represents the th root of the polynomial , where , ..., is an index indicating the root number in the Wolfram Language's ordering.The similar concept of the "th root" of a complex number is known as an nth root.The roots of a complex function can be obtained by separating it into its real and imaginary plots and plotting these curves (which are related by the Cauchy-Riemann equations)..

For two polynomials and of degrees and , respectively, the Sylvester matrix is an matrix formed by filling the matrix beginning with the upper left corner with the coefficients of , then shifting down one row and one column to the right and filling in the coefficients starting there until they hit the right side. The process is then repeated for the coefficients of .The Sylvester matrix can be implemented in the WolframLanguage as: SylvesterMatrix1[poly1_, poly2_, var_] := Function[{coeffs1, coeffs2}, With[ {l1 = Length[coeffs1], l2 = Length[coeffs2]}, Join[ NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2], NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2] ] ] ][ Reverse[CoefficientList[poly1, var]], Reverse[CoefficientList[poly2, var]] ]For example, the Sylvester matrix for and isThe determinant of the Sylvester matrix of two polynomialsis the resultant of the polynomials.SylvesterMatrix is an (undocumented)..

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