What is the most efficient way to obtain a multi-variate polynomial equation from a large data set?

My problem is as follows. I have 10 independent variables that I have permuted to arrive at about 300 different tests. Now, each of these tests give me a two-dimensional trajectory data of a partricle that is easily attained by a 6th-degree polynomical equation that relates the y-distance to x-distance. But my goal, however, is to take these 300 different 6th-degree polynomial equations (please see the plots in the image below) and incorporate with them, the 10 independent variables that I permuted to obtain these 300 different trajectories / equations. Or in other words, I would like to obtain a single grand equation that has a total of 12 variables (the 10 independent controlled ones, and the two dependent ones) that can describe (with an undertandable margin of error) all of the 300 tests. My question is two-fold: i) mathematically, how would I go about doing this? and second ii) what module in MATLAB will allow me to do this most efficiently?. I have some experience with systems identification analysis (linear time-invariant) and very primitive single input regression analysis. However, I seem to be beyond my depth with this current conundrum. My initial hunch seems to direct me towards machine learning; however, I am not sure. Thank you for your time.

[Jon]

So just to clarify, is y a function of x and the 10 independent variables, lets call them v1,v2,...v10? Does each colored curve in your plot correspond to y vs x for a particular set of values of v1,v2,...v10?

[VJ An]

Hi Jon, yes, both your questions are correct. To further clarify, not all of the 10 independent variables are changed for a single y vs. x curve. I typically change not more than two independent variables (say v1 and v2) at a time. It should also be noted that x is a function of the independent variables themselves. So, in a sense the function for one case is y=f(x,v1,v2,...v10); and x=f(v1,v2...v10). This dependency holds true for all 300 cases.