System of polynomial equations (original) (raw)
En matemáticas, un sistema de ecuaciones algebraicas es un conjunto de ecuaciones con más de una incógnita que conforman un problema matemático que consiste en encontrar los valores de las incógnitas que satisfacen dichas operaciones. Las incógnitas se suelen representar utilizando las últimas letras del alfabeto latino, o si son demasiadas, con subíndices.
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| dbo:abstract | En matemáticas, un sistema de ecuaciones algebraicas es un conjunto de ecuaciones con más de una incógnita que conforman un problema matemático que consiste en encontrar los valores de las incógnitas que satisfacen dichas operaciones. En un sistema de ecuaciones algebraicas, las incógnitas son valores numéricos menores a la constante (o más generalmente elementos de un cuerpo sobre el que se plantean las ecuaciones), mientras que en una ecuación diferencial las incógnitas son funciones o distribuciones de un cierto conjunto definido de antemano. Una solución de dicho sistema es por tanto, un valor o una función que substituida en las ecuaciones del sistema hace que éstas se cumplan automáticamente sin que se llegue a una contradicción. En otras palabras el valor que reemplazamos en las incógnitas debe hacer cumplir la igualdad del sistema. Las incógnitas se suelen representar utilizando las últimas letras del alfabeto latino, o si son demasiadas, con subíndices. (es) Sistem persamaan polinomial (terkadang hanya sistem polinomial) adalah himpunan persamaan simultan f1 = 0, ..., fh = 0 dimana fi adalah polinomial dalam beberapa variabel, misalnya x1, ..., xn, atas beberapa bidang k. Solusi dari sistem polinomial adalah sekumpulan nilai untuk xi yang termasuk dalam beberapa ekstensi bidang K dari k , dan membuat semua persamaan menjadi benar. Jika k adalah bidang bilangan rasional, K umumnya diasumsikan sebagai bidang bilangan kompleks, karena setiap solusi milik ekstensi bidang dari k , yang isomorfik ke subkolom dari bilangan kompleks. Artikel ini adalah tentang metode untuk memecahkan, yaitu menemukan semua solusi atau menjelaskannya. Karena metode ini dirancang untuk diimplementasikan di komputer, Penekanan diberikan pada bidang k yang komputasi (termasuk pengujian persamaan) mudah dan efisien, yaitu bidang bilangan rasional s dan . Mencari solusi yang termasuk dalam rangkaian tertentu merupakan masalah yang umumnya jauh lebih sulit, dan berada di luar cakupan artikel ini, kecuali untuk kasus solusi di medan hingga tertentu. Untuk kasus solusi yang semua komponennya adalah bilangan bulat atau bilangan rasional, lihat . * l * * s (in) En mathématiques, un système d'équations algébriques est un ensemble d'équations polynomiales f1 = 0, ..., fh = 0 où les fi sont des polynômes de plusieurs variables (ou indéterminées), x1, ..., xn, à coefficients pris dans un corps ou un anneau k. Une « solution » est un ensemble de valeurs à substituer aux indéterminées annulant toutes les équations du système. Généralement les solutions peuvent être cherchées dans une extension du corps k comme la clôture algébrique de ce corps (ou la clôture algébrique du corps des fractions de k celui-ci est un anneau). L'étude de l'ensemble des solutions des systèmes algébriques forme la branche des mathématiques appelée géométrie algébrique. (fr) A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k. A solution of a polynomial system is a set of values for the xis which belong to some algebraically closed field extension K of k, and make all equations true. When k is the field of rational numbers, K is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of k, which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields k in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields. Searching for solutions that belong to a specific set is a problem which is generally much more difficult, and is outside the scope of this article, except for the case of the solutions in a given finite field. For the case of solutions of which all components are integers or rational numbers, see Diophantine equation. (en) |
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| rdfs:comment | En matemáticas, un sistema de ecuaciones algebraicas es un conjunto de ecuaciones con más de una incógnita que conforman un problema matemático que consiste en encontrar los valores de las incógnitas que satisfacen dichas operaciones. Las incógnitas se suelen representar utilizando las últimas letras del alfabeto latino, o si son demasiadas, con subíndices. (es) Sistem persamaan polinomial (terkadang hanya sistem polinomial) adalah himpunan persamaan simultan f1 = 0, ..., fh = 0 dimana fi adalah polinomial dalam beberapa variabel, misalnya x1, ..., xn, atas beberapa bidang k. Solusi dari sistem polinomial adalah sekumpulan nilai untuk xi yang termasuk dalam beberapa ekstensi bidang K dari k , dan membuat semua persamaan menjadi benar. Jika k adalah bidang bilangan rasional, K umumnya diasumsikan sebagai bidang bilangan kompleks, karena setiap solusi milik ekstensi bidang dari k , yang isomorfik ke subkolom dari bilangan kompleks. (in) A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k. A solution of a polynomial system is a set of values for the xis which belong to some algebraically closed field extension K of k, and make all equations true. When k is the field of rational numbers, K is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of k, which is isomorphic to a subfield of the complex numbers. (en) En mathématiques, un système d'équations algébriques est un ensemble d'équations polynomiales f1 = 0, ..., fh = 0 où les fi sont des polynômes de plusieurs variables (ou indéterminées), x1, ..., xn, à coefficients pris dans un corps ou un anneau k. Une « solution » est un ensemble de valeurs à substituer aux indéterminées annulant toutes les équations du système. Généralement les solutions peuvent être cherchées dans une extension du corps k comme la clôture algébrique de ce corps (ou la clôture algébrique du corps des fractions de k celui-ci est un anneau). (fr) |
| rdfs:label | Sistema de ecuaciones algebraicas (es) Système d'équations algébriques (fr) Sistem persamaan polinomial (in) System of polynomial equations (en) |
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