![]() ![]() We give our solution, found at the beginning of Chapter 1 of, as it will give the flavor of our approach to this subject. ![]() The first problem for lines is to find the equation of a line through two given points. The simplest example of an algebraic plane curve is a line. We find a random line containing and use to check the point of intersection of with. (If you have set values in your session for, and so on, now is the time to store them if needed and apply to them.) įor example, suppose some calculation claims is a point on. In numerical work, we do not accept points as solutions to based on the value of, known as the residue. We will generally describe an algebraic plane curve by giving a polynomial in two variables with integer or real machine-number coefficients.įrom an operational point of view, with an exception noted later, for a given curve we accept the output of NSolve, where is another such bivariate polynomial and, are real machine numbers, as points on the curve. Having worked during my career as a mathematician in both the abstract and numerical realms, I believe that while these approaches are incompatible, they can and should coexist within mathematics. So my book is not an algebraic geometry book. Evaluating this polynomial at a point with machine-number coordinates gives a machine number on the left-hand side, while the right-hand side is a symbolic number, so actual equality is impossible. In fact there is a fundamental oxymoron at the heart of my approach: a numerical algebraic curve is the solution set of an equation, where is a polynomial with integer or machine-number coefficients. The methods are constructive, heuristic and visual rather than the traditional theorem-proof of contemporary mathematics. Since most algebraic curves have only finitely many rational points, I work numerically. One goal of my book is to rectify this problem substituting software for the abstract theory, we can give the theory in terms the non-mathematician can follow. In addition, little attention was given to the concrete geometric theory. All modern books on the subject want to follow the abstract approach, which raises the bar for those who want to know this theory. The plane geometric curve theory of the nineteenth century was collateral damage. Probably the most striking accomplishment of this abstract approach was the solution of Fermat ’s problem by Wiles and Taylor at the end of the century. The added benefit of this approach is that it became possible to apply geometric techniques to other fields. Ideas of ideals, rings, fields, varieties, divisors, characters, sheaves, schemes and many types of homology and cohomology arose. In the absence of an ability to do the large number of computations for a concrete theory, the twentieth century saw the abstraction to algebraic geometry of this material. The nineteenth century saw great progress in geometric (real) and analytic (complex) algebraic plane curves. This article is a summary of my book A Numerical Approach to Real Algebraic Curves with the Wolfram Language. ![]()
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