Polynomial Basis Functions

The basis is not one polynomial chosen from a single exponent pattern. It is the complete collection produced by combining all allowed exponent choices.

Programming

From State to Functions

Suppose a state has d real-valued variables. An OrderN polynomial basis converts that state into a collection of polynomial basis functions. Each function is formed by choosing one allowed integer exponent for every state variable, raising each variable to its chosen exponent, and multiplying the resulting terms together.

Key point

The basis is not one polynomial chosen from a single exponent pattern. It is the complete collection produced by combining all allowed exponent choices.

How do the state variables combine with their exponents to form one polynomial basis function?

Exponent Choices

For every state variable, its corresponding exponent may be any integer from 0 through N. The notation c_i,j identifies the exponent assigned to a particular state variable in a particular basis function: the first index identifies the variable position, and the second identifies the basis-function choice. Thus, for a state with variables s₁ and s₂, one exponent controls the power of s₁ and another controls the power of s₂.

Changing an exponent changes the term contributed by its variable. For example, an exponent pair of (1, 2) produces s₁s₂². An exponent pair of (2, 0) produces s₁². The second variable does not appear as a factor in the written product because its power is zero.

Which exponent belongs to each state variable, and how does changing an exponent change the resulting term?

Multiplying the Terms

Constructing one basis function follows the same pattern for every number of state variables. First choose an allowed exponent for the first variable and form its powered term. Repeat this for the second variable and every remaining variable. Finally, multiply one powered term for every state variable. The result is one polynomial basis function.

How does each powered state-variable term contribute to the final product that defines a basis function?

Worked example

One Function from Three Variables

Choose exponents 2, 0, and 1 for state variables s₁, s₂, and s₃.

1
Assign the powers
The exponent for s₁ is 2, the exponent for s₂ is 0, and the exponent for s₃ is 1.
2
Form the powered terms
The terms are s₁², s₂⁰, and s₃¹.
3
Multiply the terms
Multiplying one powered term for each variable gives s₁²s₂⁰s₃¹, which can be written as s₁²s₃.

Result

The chosen exponent combination produces the basis function s₁²s₃.

Counting the Complete Basis

Each of the d state variables has N + 1 possible exponent choices: 0, 1, and every integer through N. A complete basis function requires one choice for each variable. Because every allowed choice is combined with every allowed choice for the other variables, the complete OrderN basis contains (N + 1)^{^d} different functions.

How do the number of state variables d and the order N determine how many basis functions are included?

Worked example

Counting a Two-Variable Order-2 Basis

Find the number of functions when d = 2 and N = 2.

1
Count choices for one variable
Each exponent can be 0, 1, or 2, so there are N + 1 = 3 choices.
2
Combine the choices
There are 3 choices for the first variable and 3 choices for the second variable.
3
Apply the basis count
The number of functions is (N + 1)^{^d} = 3^{^2} = 9.

Result

A two-variable Order-2 polynomial basis contains 9 functions.

A Complete Small Basis

Take two state variables, s₁ and s₂, and choose Order 2. The possible exponent values for each variable are 0, 1, and 2. Represent each function by an exponent pair: the first entry controls s₁ and the second controls s₂.

What are all the individual basis functions when two state variables and Order 2 are chosen?

Exponent pair	Basis function
(0, 0)	1
(0, 1)	s₂
(0, 2)	s₂²
(1, 0)	s₁
(1, 1)	s₁s₂
(1, 2)	s₁s₂²
(2, 0)	s₁²
(2, 1)	s₁²s₂
(2, 2)	s₁²s₂²

Every pair chooses one exponent for s₁ and one exponent for s₂.

Mistakes to Avoid

Common mistake
Using only exponents 1 through N
What it looks like
Leaving out exponent 0 when N = 2.
Why it fails
Each exponent may be any integer from 0 through N, so omitting 0 removes valid basis functions such as 1, s₁, or s₂².
Do this instead
List 0, 1, and every integer through N before combining choices.
Common mistake
Assigning both exponents to the same variable
What it looks like
```
Treating the pair (1, 2) as s₁³.
```
Why it fails
The first exponent controls s₁ and the second controls s₂.
Do this instead
Read each position in the exponent choice as belonging to its corresponding state variable; (1, 2) produces s₁s₂².
Common mistake
Counting only the individual exponent values
What it looks like
For d = 2 and N = 2, counting 3 functions instead of 9.
Why it fails
A function requires one exponent choice for every variable, and all choices are combined.
Do this instead
Use (N + 1)^d. Here, 3 choices are made twice, giving 3² = 9.
Common mistake
Forgetting a variable with exponent zero
What it looks like
Rejecting s₁² as incomplete because s₂ is not visibly present.
Why it fails
The exponent 0 is allowed, so s₁² represents the pair (2, 0).
Do this instead
Treat a zero exponent as a valid choice for that variable.

Practice Construction

EASY

Construct the complete polynomial basis for d = 2 and N = 1. List every exponent pair and write the corresponding basis function. Then determine the number of functions using (N + 1)^{^d}.

Hints

Each exponent can be 0 or 1.
Pair the first exponent with s₁ and the second exponent with s₂.
There should be (1 + 1)^{^2} functions.

Your attempt (not saved)

MEDIUM

A state has d = 3 variables and uses Order 2. How many functions are in the complete basis? Explain what the three exponent positions control in one chosen function.