By nonlinear algebra I mean things relating to multivariate polynomials

Linear modelling of nonlinear systems

Vandermonde Matrix

The big question: Linear in what? You can always determine a polynomial as a linear algebra problem given sample points. A polynomial can bethought of as linear in it’s coefficients

Companion Matrix

wiki A matrix designed to have characterstic polynomial

Sylvester matrix

Univariate Polynomials

We’re really good at these. Fundamental theorem of algebra

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Homotopy Continuation

Grobner Bases

Cylindrical Algebraic Decomposition (CAD)

NLSAT

Modules

Linear algebra but wih rings instead of fields. You can’t divide by scalars anymore.

Resources

https://github.com/algebraic-solving/msolve msolve. a C library for grobner basis and root calculations. Intresting