I am working on soil biology. I have a data set of soil CO2 emissions (dependant variable), with a number of other measured variables (independents).

I expect that one of the independant variables (A) is a straight linear relationship. But expect that another variable (B), is some kind of polynomial shape, probably quadratic or something.

Is there some kind of analysis i could do to sum up these relationships?

Additional info: I have already done a linear mulitple regression, it correlates variable A significantly, but not B. But playing around with it, i have found that if i split the dataset in half around the limit of the curve for variable B, then run multiple regressions on both partial data sets, i get significant linear corrleations for variable B (one is positive, one is negative. See the picture which is a rough drawing of what i have.

I have a number of other variables too, they dont correlate significantly but i want to include them anyway because it completes the story.