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In Chapter 1, we have seen how the algebra of the polynomial rings k[x1, . . . , xn] and the geometry of affine varieties are linked. In this chapter, we will study the method of Gröbner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. The method of Gröbner bases is also used in several powerful computer algebra systems to study specific polynomial ideals that arise in applications. In Chapter 1, we posed many problems concerning the algebra of polynomial ideals and the geometry of affine varieties. In this chapter and the next, we will focus on four of these problems.

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Gröbner Bases

  • 章节

  • 发表日期：

    2025年

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  • David A. Cox,
  • John Little,
  • Donal O’Shea

摘要

In Chapter 1, we have seen how the algebra of the polynomial rings k[x1, . . . , xn] and the geometry of affine varieties are linked. In this chapter, we will study the method of Gröbner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. The method of Gröbner bases is also used in several powerful computer algebra systems to study specific polynomial ideals that arise in applications. In Chapter 1, we posed many problems concerning the algebra of polynomial ideals and the geometry of affine varieties. In this chapter and the next, we will focus on four of these problems.