MathObj.LaurentPolynomial

Polynomial including negative exponents

Add lists of numbers respecting a relative shift between the starts of the lists. The shifts must be non-negative. The list of relative shifts is one element shorter than the list of summands. Infinitely many summands are permitted, provided that runs of zero shifts are all finite.

We could add the lists either with foldl or with foldr, foldl would be straightforward, but more time consuming (quadratic time) whereas foldr is not so obvious but needs only linear time.

(stars denote the coefficients, frames denote what is contained in the interim results) foldl sums this way:

| | | *******************************
| | +--------------------------------
| |          ************************
| +----------------------------------
|                        ************
+------------------------------------

I.e. foldl would use much time find the time differences by successive subtraction 1.

foldr mixes this way:

    +--------------------------------
    | *******************************
    |      +-------------------------
    |      | ************************
    |      |           +-------------
    |      |           | ************

Two polynomials may be stored differently. This function checks whether two values of type LaurentPolynomial actually represent the same polynomial.

Check whether a Laurent polynomial has only the absolute term, that is, it represents the constant polynomial.

p(z) -> p(-z)

p(z) -> p(1/z)

p(exp(i·x)) -> conjugate(p(exp(i·x)))

If you interpret (p*) as a linear operator on the space of Laurent polynomials, then (adjoint p *) is the adjoint operator.