TITLE: How to implement Horner’s scheme for multivariate polynomials?
QUESTION [9 upvotes]: Background
I need to solve polynomials in multiple variables using Horner's scheme in Fortran90/95. The main reason for doing this is the increased efficiency and accuracy that occurs when using Horner's scheme to evaluate polynomials.
I currently have an implementation of Horner's scheme for univariate/single variable polynomials. However, developing a function to evaluate multivariate polynomials using Horner's scheme is proving to be beyond me.

An example bivariate polynomial would be: $12x^2y^2+8x^2y+6xy^2+4xy+2x+2y$ which would factorised to $x(x(y(12y+8))+y(6y+4)+2)+2y$ and then evaluated for particular values of x & y.

Research
I've done my research and found a number of papers such as:
staff.ustc.edu.cn/~xinmao/ISSAC05/pages/bulletins/articles/147/hornercorrected.pdf
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.8637&rep=rep1&type=pdf
www.is.titech.ac.jp/~kojima/articles/B-433.pdf  
Problem
However, I'm not a mathematician or computer scientist, so I'm having trouble with the mathematics used to convey the algorithms and ideas.
As far as I can tell the basic strategy is to turn a multivariate polynomial into separate univariate polynomials and compute it that way.
Can anyone help me? If anyone could help me turn the algorithms into pseudo-code that I can implement into Fortran myself, I would be very grateful.

REPLY [6 votes]: I implemented this in Python: multivar_horner
You can look at the approach used there and port it to Fortran.
NOTE: In contrast to the one dimensional case there are multiple possible Horner factorisations of multivariate polynomials. One can allow a search over the possible factorisations to find a minimal representation as described HERE. In most cases however it suffices to use a heuristic to find a single "good" factorisation. multivar_horner implements the greedy heuristic described in "Greedy Algorithms for Optimizing Multivariate Horner Schemes".
The authors of some related publications (including the above mentioned) claim to have an implementation of their proposed algorithms, but I was not able to find any publicly available ones.