4. Multivariate: Base Multinumbers

To go beyond one variable, Wildberger introduces base multinumbers — a family of primitive elements, one for each natural number index:

$$e_0,\; e_1,\; e_2,\; e_3,\;\ldots$$

Each $e_n$ is just a tag — a box holding the natural number $n$. Polynomials in multiple variables are formed by taking products and sums of these tags with natural number coefficients. So where ordinary algebra writes $x$ and $y$, box arithmetic writes $e_0$ and $e_1$ (or any two distinct indices).

A polynumber term in several variables is a product of base multinumbers:

$$e_1^2 \cdot e_3 = e_1 e_1 e_3$$

In boxmath, exponents[i] is the power of $e_i$:

Polynumber	Exponent vector
$1$ (constant)	[]
$e_1$	[0, 1]
$e_1^2$	[0, 2]
$e_2 e_4$	[0, 0, 1, 0, 1]
$e_1^2 e_3$	[0, 2, 0, 1]

Coefficients are just the natural number scalar

new Polynumber(2n, [0,0,0,1]) is $2e_3$ — the exponent vector says which variables appear and at what power; the first argument is the natural number coefficient. So $1 + 2e_3 + e_2 e_4$ encodes as:

new Multinumber([
  new Polynumber(1n, []),          // 1
  new Polynumber(2n, [0,0,0,1]),   // 2e₃
  new Polynumber(1n, [0,0,1,0,1]), // e₂e₄
]);

Trailing zeros can always be dropped

Wildberger states explicitly: "adding or removing final 0 entries does not change the representation." The reason is that $e_i^0 = 1$ for any $i$ — an exponent of zero contributes nothing to the product. So [0,0,0,1] and [0,0,0,1,0] are the same polynumber term ($e_3$). The library respects this: Polynumber.evaluate only multiplies when exponents[i] > 0, and Polynumber.extent finds the last nonzero index and ignores everything after it.

Wildberger's worked example (PDF §3.5)

"If $B = 1 + e_3 + e_2 e_4$ and $C = e_1^2 + e_2 e_4$ then $B \cdot C = e_1^2 + e_2 e_4 + e_1^2 e_3 + e_2 e_3 e_4 + e_1^2 e_2 e_4 + e_2^2 e_4^2$."

The product is every pairwise combination of a term from $B$ with a term from $C$ — $3 \times 2 = 6$ terms total:

import { Polynumber, Multinumber } from 'boxmath';

const B = new Multinumber([
  new Polynumber(1n, []),          // 1
  new Polynumber(1n, [0,0,0,1]),   // e₃
  new Polynumber(1n, [0,0,1,0,1]), // e₂e₄
]);

const C = new Multinumber([
  new Polynumber(1n, [0,2]),       // e₁²
  new Polynumber(1n, [0,0,1,0,1]), // e₂e₄
]);

const BC = B.multiply(C);
BC.terms.length;  // 6

BC.terms.map(t => t.toString(['e0','e1','e2','e3','e4']));
// ['e1^2', 'e2e4', 'e1^2e3', 'e2e3e4', 'e1^2e2e4', 'e2^2e4^2']

The exponent vectors add component-wise under multiply — the box product rule.