Multilinear Functions and Polynomials

In summary, a multilinear function is a mathematical function with multiple variables raised to a power of one, while a polynomial consists of one or more terms with variables raised to non-negative integer powers. They can both be represented graphically on a coordinate plane and have a degree that represents the highest power of the variable. These types of functions are commonly used in fields such as physics, engineering, and economics to model relationships between variables. Common operations performed on them include addition, subtraction, multiplication, and division to simplify or manipulate the functions.
  • #1
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A function [tex]f : \mathbf{R}^n\rightarrow\mathbf{R}[/tex] is multilinear if it's linear in every variable. Is there a multilinear function that's not a multilinear polynomial?

Given a function defined on the n dimensional hypercube, values of which are 0 or 1, there is a unique multilinear extension to all of R. Is this extension a polynomial?
 
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  • #2
No, a "linear function" is a polynomial, by definition, so any multilinear function is a polynomial. Saying "multilinear polynomial" is redundant.
 

1. What is the difference between a multilinear function and a polynomial?

A multilinear function is a mathematical function that involves multiple variables, each with a power of one. On the other hand, a polynomial is a mathematical expression that consists of one or more terms, with each term containing a variable raised to a non-negative integer power. Therefore, all polynomials are multilinear functions, but not all multilinear functions are polynomials.

2. How can multilinear functions and polynomials be represented graphically?

Both multilinear functions and polynomials can be represented graphically by plotting points on a coordinate plane. The x-axis represents the input variables, and the y-axis represents the output values. The resulting graph will show the relationship between the variables and the function's output.

3. What is the degree of a multilinear function or polynomial?

The degree of a multilinear function or polynomial is the highest power of the variable in the function or polynomial. For example, a multilinear function with variables x, y, and z, each with a power of 1, would have a degree of 1. A polynomial with a term of 5x^3 would have a degree of 3.

4. How are multilinear functions and polynomials used in real life?

Multilinear functions and polynomials are used in various fields, including physics, engineering, and economics, to model relationships between multiple variables. For example, a multilinear function can be used to calculate the force of an object based on its mass, acceleration, and friction. Polynomials can be used to model the growth of a population over time or the revenue of a business based on its sales and expenses.

5. What are some common operations performed on multilinear functions and polynomials?

Some common operations performed on multilinear functions and polynomials include addition, subtraction, multiplication, and division. These operations are used to simplify, solve, or manipulate the functions and polynomials to better understand their relationships and properties.

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