Example of basis for d-variate polynomials for numerical application

I want to construct a polynomial basis for the space of $$d-$$variate polynomials of degree $$Q$$ for numerical purposes.

Let's pick for example $$Q = 2$$ and $$d = 3$$.

Now, if $$d = 1$$ it would be easy: I would just pick: $${1,x,x^2}$$ they are linearly independent and span the space of polynomials of max degree 2. However if d = 3 then I am looking for polynomials of the form: $$p_1(x,y,z),p_2(x,y,z),p_3(x,y,z),...,$$

I found that I would need $$Q+d$$ polynomials to have a basis, is that true? Moreover, my application is a numerical one, so I would need a basis that works well numerically, like for example Chebyshev polynomials: they work well in 1D.

Can I get some pointers as where to look?

EDIT: Just to be more clear, in 1D on an interval [a,b] I have the following "formula" for Chebyshev polynomials: $$Cheb(x,k) = \cos{(k*\arccos{[-1 + \frac{2}{(b-a)}*(x-a)]})}$$

Which gives me the value at the point x of the k-th Chebyshev polynomial. I was wondering if there existed something similar for degree other than one.

EDIT 2: I also heard about centralized and scaled monomials of the form $$p_\alpha(x)=(\frac{x-c}{d})^\alpha$$ where d is half diameter of the domain specified and c is the "center" of the domain. In dimension 1 it is pretty clear.

My question then is: in dimension $$d = 3$$ can this idea be generalized as: $$p_\alpha(x,y,z) = (\frac{xyz- \Vert c \Vert}{d})^\alpha$$

• There are $\binom{Q+d}{d}$ monomials of degree $Q$ or smaller in $d$ variables. Think of a peg-hole situation, $Q+d$ holes and $d$ pegs, the number of holes in segment $i$ before peg $i$ gives the degree of the $i$-th variable in the monomial. There can be holes left after the last peg for monomials of non-maximal degree. – LutzL May 31 at 11:52