6.1 Symbolic methods for solving polynomial systems

Symbolic methods for solving polynomial systems are powerful tools in computational algebraic geometry. They transform complex polynomial equations into simpler forms, making them easier to solve. Gröbner bases and multivariate polynomial division are key techniques in this approach.

These methods can be computationally intensive, especially for large systems. However, they're invaluable for solving zero-dimensional systems and can be optimized through careful choice of monomial ordering and leveraging system structure. Understanding their complexity is crucial for effective application.

Gröbner Basis Methods

Fundamentals of Gröbner Bases

• Gröbner bases are special sets of polynomials associated with a given ideal in a polynomial ring that have useful algorithmic properties for solving systems of polynomial equations
• The main method for computing Gröbner bases is Buchberger's algorithm, which relies on a generalization of polynomial long division in multiple variables
• Gröbner bases allow transforming the problem of solving polynomial systems into the problem of solving triangular systems, which can be solved more easily by back-substitution
• The elimination property of Gröbner bases states that the intersection of the ideal with lower-dimensional polynomial subrings can be found by taking certain subsets of the basis, which is key for solving polynomial systems

Complexity and Suitability of Gröbner Bases

• The choice of monomial ordering in Gröbner basis computation greatly influences the form of the basis and the efficiency of the algorithm with common orderings including lexicographic, graded lexicographic, and graded reverse lexicographic order
• The complexity of Gröbner basis computation can be doubly exponential in the number of variables in worst case, and the choice of monomial ordering and polynomial structure impact practical efficiency
• Gröbner bases are well-suited for solving zero-dimensional polynomial systems that have finitely many solutions
• Positive-dimensional systems may require additional techniques like primary decomposition to be solved using Gröbner bases

Multivariate Polynomial Division

Multivariate Division Algorithms

• Multivariate division algorithms like multivariate long division and reduction modulo a Gröbner basis generalize polynomial long division to the case of multiple variables
• Multivariate long division of a polynomial $f$ by a set of polynomials $F = {f_1,\ldots,f_s}$ produces a representation $f = \sum a_i f_i + r$ where $r$ is a "remainder" that cannot be further divided by the elements of $F$
• Reduction modulo a Gröbner basis $G = {g_1,\ldots,g_t}$ of an ideal $I$ allows writing any polynomial $f$ as $f = \sum b_i g_i + \overline{f}$ where $\overline{f}$ is a unique remainder called the normal form of $f$ modulo $I$

Applications of Polynomial Division

• Reduction modulo a Gröbner basis can be used to test ideal membership - $f \in I$ if and only if its normal form modulo $I$ is zero
• Polynomial division and reduction are the key operations underlying Buchberger's algorithm for computing Gröbner bases
• To solve a polynomial system $f_1=\cdots=f_s=0$, first compute a Gröbner basis $G$ of the ideal $\langle f_1,\ldots,f_s \rangle$, then use the elimination property and reduction to triangularize and back-substitute

Complexity of Symbolic Algorithms

Complexity Measures

• The complexity of Gröbner basis algorithms is usually measured in terms of the Hilbert function or Hilbert polynomial of the underlying ideal, which describe the size of the quotient ring
• In worst case, the degrees of polynomials during Gröbner basis computation can grow doubly exponentially in the number of variables, however, this is not typical for most systems in practice
• The most expensive operation in Buchberger's algorithm is the computation of S-polynomials, whose number grows quadratically with the size of the basis, but criteria exist to avoid some S-polynomial reductions

Algorithmic Improvements

• Faugère's F4 and F5 algorithms improve on Buchberger by computing fewer S-polynomials and using linear algebra to perform reductions more efficiently
• Complexity of Gröbner basis computation is a benchmark for comparing symbolic algorithms, with other important measures being memory usage and performance on classes of "structured" systems
• Gröbner bases can be computed more efficiently if the system is known to have certain properties like sparsity, homogeneity, or specific elimination structure, and algorithms should leverage this when possible
• Modular and parallel computation techniques can be used to improve efficiency of Gröbner basis algorithms on large systems

Term Ordering Schemes

Monomial Ordering Fundamentals

• Implementing Gröbner basis algorithms requires efficient data structures and algorithms for working with polynomials and monomials, e.g. distributed representations, linked lists, heaps
• Monomial orderings are defined as well-orderings on the set of monomials that are compatible with multiplication and they determine how polynomials are represented and which terms are eliminated first
• Lexicographic order (lex) eliminates variables one at a time and results in "triangular" bases, but often causes explosive intermediate expression swell while graded orders like grevlex can help control this
• Graded reverse lexicographic order (grevlex) first compares total degrees and then does reverse lex on variables, typically producing more compact bases and serving as a good default choice

Advanced Ordering Techniques

• Product or block orders combine different orderings for subsets of variables, e.g. eliminate some variables with lex and others with grevlex, and they can be used to compute elimination ideals more efficiently
• Weighted orderings assign different weights to variables and compare monomials by weighted degree, allowing prioritization of elimination of certain variables or control over the shape of the basis
• Matrix term orderings represent monomials as matrices and compare them based on their Smith normal form, exhibiting useful geometric and homological properties
• Implementing Gröbner bases requires choosing appropriate monomial orderings for the problem at hand and optimizing data structures for those orderings, with experimentation of different orderings potentially greatly impacting performance