4.5 Standard form structure

A standard form is a polynomial in internal recursive representation where the kernel order defines the recursive structure. E.g. with kernel order (x y) the polynomial x²y + x² + 2xy + y² + x + 3 is represented as a polynomial in x with coefficients which are polynomials in y and integer coefficients: x² * (y * 1 + 1) + x * (y * 2 + 1) + (y² * 1 + 3); for better correspondence with the internal representation here the integer coefficients are in the trailing position and the trivial coefficients 1 are included. A standard form is

• <domain element>
• <mvar> .** <ldeg> .* <lc> .+ <red>, internally represented as (((mvar.ldeg).lc).red)

with the components

• mvar: the leading kernel,
• ldeg: an integer representing the power of mvar in the leading term,
• lc: the leading coefficient is itself a standard form, not containing mvar any more,
• red: the reductum too is a standard form, containing mvar at most in powers below ldeg.

Note that any standard form ends with a domain element which is nil if there is no constant term. E.g. the above polynomial will be represented internally by

with the components

• mvar: X
• ldeg: 2
• lc: ((Y. 1). 1). 1) = (y + 1)
• red: (((X. 1) ((Y. 1). 2)) ((Y. 2). 1). 3) = xy² + 3.