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# DecomposableSparseSystems -- Solving decomposable sparse polynomial systems

## Description

There are two natural ways a sparse polynomial system can be decomposed in the sense of (T. Brysiewicz, J. I. Rodriguez, F. Sottile, and T. Yahl, Solving Decomposable Sparse Systems, arXiv:2001.04228, 2019). These methods detect and compute these decompositions and use them to compute solutions.

## Certification

Version 1.0.1 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 25 March 2021, in the article Decomposable sparse polynomial systems. That version can be obtained from the journal or from the Macaulay2 source code repository.

## Version

This documentation describes version 1.0.1 of DecomposableSparseSystems.

## Source code

The source code from which this documentation is derived is in the file DecomposableSparseSystems.m2. The auxiliary files accompanying it are in the directory DecomposableSparseSystems/.

## Exports

• Functions and commands
• isDecomposable -- Decides whether a polynomial system is decomposable
• isLacunary -- Decides whether a polynomial system is lacunary
• isTriangular -- Decides whether a polynomial system is triangular
• solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• Methods
• "isDecomposable(List)" -- see isDecomposable -- Decides whether a polynomial system is decomposable
• "isLacunary(List)" -- see isLacunary -- Decides whether a polynomial system is lacunary
• "isTriangular(List)" -- see isTriangular -- Decides whether a polynomial system is triangular
• "solveDecomposableSystem(List)" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• "solveDecomposableSystem(List,List)" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• "solveDecomposableSystem(List,Nothing)" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• Symbols
• "FromGeneric" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• "FromVertexGeneric" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• "LacunarySystem" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition
• "TriangularSystem" -- see solveDecomposableSystem -- recursively solves a sparse (Laurent) polynomial system through a decomposition

## For the programmer

The object DecomposableSparseSystems is .