5.3.5 Poly

The general object of this type of degree $2\leq$ORDER$\leq 15$ is the poly object with syntax:

POLY:
  poly { ORDER, < POLY_COEFFICIENTS > [POLY_MODIFIERS] }

POLY_COEFFICIENTS:
  A quantity n of FLOATs separated by commas, 
    where n is ((ORDER+1)*(ORDER+2)*(ORDER+3))/6.

POLY_MODIFIERS:
  [sturm [BOOL]] & [OBJECT_MODIFIERS]

See also:

• [pov:6.5.3.2]Poly, Cubic and Quartic
• [pov:3.6.4]Poly Object

It describes the object given by

$$\Bigl\{\langle X,Y,Z\rangle: \sum_{i=0}^d X^{d-i} \sum_{j=0}^{i} Y^{i-j}\sum_{k=0}^{j} Z^{j-k}\,a_{i,j,k} \leq 0 \Bigr\}$$

where the coefficients are given in exactly this order. Note that this is not the usual ordering by degree of homogeneity. But the inverse lexicographical ordering of the sequence of exponents of $X$, $Y$ and $Z$. Thus for degree 3 the 20 monomials are ordered as

$$X^3 Y^0 Z^0$$

$$X^2 Y^1 Z^0 \quad X^2 Y^0 Z^1 \quad X^2 Y^0 Z^0$$

$$X^1 Y^2 Z^0 \quad X^1 Y^1 Z^1 \quad X^1 Y^1 Z^0 \quad X^1 Y^0 Z^2 \quad X^1 Y^0 Z^1 \quad X^1 Y^0 Z^0$$

$$X^0 Y^3 Z^0 \quad X^0 Y^2 Z^1 \quad X^0 Y^2 Z^0 \quad X^0 Y^1 Z^2 \quad X^0 Y^1 Z^1 \quad X^0 Y^1 Z^0$$

$$\qquad X^0 Y^0 Z^3 \quad X^0 Y^0 Z^2 \quad X^0 Y^0 Z^1 \quad X^0 Y^0 Z^0$$

The number of coefficients of a polynomial of degree at most ORDER($=d$) in the three variables $X$, $Y$ and $Z$ has as many coefficients, as there are triples $(i,j,k)$ with $i+j+k\leq d$ with $i,j,k\geq 0$. Such triple can be equally described by three numbers $0\leq k\leq j\leq i\leq d$ as in the sum above. Such 3 numbers can be viewed as 3 separators at 3 different positions among $d+3$ many items, where $k$ is the number of non-separators to the left of the first one, $j$ is that of non-separators to the left of the second and $i$ is that of non-separators to the left of the third one. By combinatorics the number of such choices is

$$\binom{n+3}{3}=\frac{(n+3)(n+2)(n+1)}{6}.$$