Background
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I need to solve polynomials in multiple variables using [Horner's scheme](http://en.wikipedia.org/wiki/Horner_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.



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> 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.
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Research
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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
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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.



How to implement Horner’s scheme for multivariate polynomials?
The paper you cite, "On the multivariate Horner scheme" (Pena, Sauer) has an explicit algorithm specified on p.3. The remaining challenge is to penetrate the notation and conventions in the paper
laid out in the first three pages far enough to turn their algorithm presentation into code.


It also seems that this paper (just reading the abstract) specifies an explicit algorithm:
"Evaluation of Multivariate Polynomials and Their Derivatives,"
J. Carnicer and M. Gasca,
*Mathematics of Computation*, Vol. 54, No. 189 (Jan., 1990), pp. 231-243.
[ResearchGate link to full text](https://www.researchgate.net/publication/258986835_Evaluation_of_Multivariate_Polynomials_and_Their_Derivatives).



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> **Abstract**. An extension of Horner's algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.
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