abtract taylor series Maple

Question

I would like to solve the folowing equation at some finite order: $$\langle f_x,f_y\rangle=0, \vert f_x\vert^2=\vert f_y\vert^2$$ where $f: \mathbb{R}^2 \rightarrow\mathbb{R}^3$.

In order to do so, I write a formal taylor series

$$f(x,y)=x<1,0,0>+y<0,1,0>+x^2<a,b,c>+2x*y<d,e,f>+y^2<g,h,i>+\dots$$ Then I plug it into the above equation and I get conditions on $a,b,c,\dots$.

The Problem is that I would like to push to the 6 order. How can I defined the serie abstractly, in order that Maple generates automatically the letter. I mean I am looking for a function like "abstract taylor" which will gives for instance

$$ Abstracttaylor(3,[x,y])= a+b*x+c*y+d*x^2+2*x*y*e+*y^2 $$

Answer 1

There is no command in Maple for this, but it is possible to generate an abstract polynomial with unknown coefficients using something like:

Abstractbivariatepolynomial := proc(d::nonnegint, v::[name,name], a::name)
    local i, j;
    return add(add(a[i,j]*x^i*y^j,j=0..d-i),i=0..d);
end proc:

Abstractbivariatepolynomial(3,[x,y],a);