Methods in Algebraic Geometry towards Data Science Applications
Intro
- Algebraic geometry deals with non-linar objects(Polynomials)
- more computationally expensive
algebraic geometry
- study the 0s of polynomials
- \(Z(f):=\{X \) such that \(f(X) = 0\}\)
| Polynomial | Zeros |
|---|---|
| f(x) = x - 7 | Z(f) = \{7\} |
| \(f_1(x,y) = y-x\) | line through the origin |
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Question
- Given a set of polynomials, find all the common zeros
- Gaussien elimination is used in the linear case
Grobner Bases
- 2 equations, two variables
Gausiien alimination
- Elimiminate leading term for every pair of linear equations
- Continue until you reach and equation with just one variable
- Solve system with back substitution
Two polynomials
- \(x^2y^3 - 4 = 0\)
- \(x^3y^2 - 2 = 0\)
- multiply first polynomial by x, second by y
- \(x^3y^3 - 4x = 0\)
- \(x^3y^3 - 4y = 0\)
- subtract the equations
- \(y = 2x\)
- plug into original polynomial
- \(x = 1\)
generalized gaussian
- modified gaussian works in the non-linear case
- S-polynomial of f and g is S(f,g)
- compute every S polynomial of every possible pair of polynomials until system can be solved
- set of all S-polynomials is called a Grobner Basis
Buchberger's Algorithm for Grobner Basis
- iterate over every pair of polynomials in S
- compute S(f,G) and add it to S'
- repeat until S is solvable
Geometrity of gaussian elimination
Geometry of Grobner Basis
Applications of Grobner Bases
- S is not solvable if and only if the grobner basis contains a constant polynomial
- This is show gemetrically by intersections in the zero polynomials
Sudoku
- Must contain all numbers 1,4 exactly once
- All columns and rows must add up to 10
- Use grobner basis to reduce polynomial to solvable state
Conclusion
- Grobner bases are a very power notion
- many applications (automated geometry theorem, robotics, sudoku, etc)
- critical insight into any system of polynomials
- lots of software can compute grobner bases