Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

13Pxx Computational commutative algebra

This subtopic studies computational commutative algebra, focusing on algorithms for Gröbner bases, ideals, and algebraic systems arising in practice.

Specific topics

13P05 Polynomials, factorization

Overview

13P05 studies polynomials, factorization in computational commutative algebra. It focuses on algorithmic representations, complexity-sensitive methods, and symbolic workflows used to make commutative-algebra problems explicit and computable.

Related Wikipedia Page

Polynomials, factorization (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for polynomials, factorization
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Symbolic computation in algebraic geometry
  • Algorithm design for ideals and modules
  • Applications in coding, robotics, and data-driven modeling

References

Recommended Textbooks

13P10 Gröbner bases; other bases for ideals and modules

Overview

13P10 studies grobner bases; other bases for ideals and modules in computational commutative algebra. It focuses on algorithmic representations, complexity-sensitive methods, and symbolic workflows used to make commutative-algebra problems explicit and computable.

Related Wikipedia Page

Grobner bases; other bases for ideals and modules (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for grobner bases; other bases for ideals and modules
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Symbolic computation in algebraic geometry
  • Algorithm design for ideals and modules
  • Applications in coding, robotics, and data-driven modeling

References

Recommended Textbooks

13P15 Solving polynomial systems

Overview

13P15 studies solving polynomial systems in computational commutative algebra. It focuses on algorithmic representations, complexity-sensitive methods, and symbolic workflows used to make commutative-algebra problems explicit and computable.

Related Wikipedia Page

Solving polynomial systems (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for solving polynomial systems
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Symbolic computation in algebraic geometry
  • Algorithm design for ideals and modules
  • Applications in coding, robotics, and data-driven modeling

References

Recommended Textbooks

13P20 Computational homological algebra

Overview

13P20 studies computational homological algebra in computational commutative algebra. It focuses on algorithmic representations, complexity-sensitive methods, and symbolic workflows used to make commutative-algebra problems explicit and computable.

Related Wikipedia Page

Computational homological algebra (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for computational homological algebra
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Symbolic computation in algebraic geometry
  • Algorithm design for ideals and modules
  • Applications in coding, robotics, and data-driven modeling

References

Recommended Textbooks

13P25 Applications of commutative algebra to sciences

Overview

13P25 studies applications of commutative algebra to sciences in computational commutative algebra. It focuses on algorithmic representations, complexity-sensitive methods, and symbolic workflows used to make commutative-algebra problems explicit and computable.

Related Wikipedia Page

Applications of commutative algebra to sciences (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for applications of commutative algebra to sciences
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Symbolic computation in algebraic geometry
  • Algorithm design for ideals and modules
  • Applications in coding, robotics, and data-driven modeling

References

Recommended Textbooks