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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

• 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

(((X .2) ((Y. 1). 1). 1) ((X. 1) ((Y. 1). 2)) ((Y. 2). 1). 3)

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.

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