Positive polynomial

In mathematics, a positive polynomial is a special kind of math expression. It always gives a positive result when we use numbers from a certain group. A non-negative polynomial always gives a result that is zero or higher, never negative.

Polynomials are expressions made from variables and numbers, like ( x^2 + 2x + 1 ). When we say a polynomial is positive on a set, it means every number from that set will give a result greater than zero. This idea is useful in many areas of math. It helps solve problems and prove theories.

These polynomials are studied in Euclidean space. This is the space we usually think of when we talk about points with coordinates, like on a graph. Understanding positive and non-negative polynomials helps experts in fields like optimization and algebraic geometry. For more advanced ideas, see the Krivine–Stengle Positivstellensatz.

Positivstellensatz and nichtnegativstellensatz

Some special groups have rules that help us describe all polynomials that stay positive or non-negative on them. These rules are called positivstellensatz for positive polynomials and nichtnegativstellensatz for non-negative ones. These ideas are important in computing because they can change hard math problems into easier ones that we can solve quickly using special methods. In certain cases, these problems become even simpler, focusing on finding big numbers from specific matrices.

Main article: polynomial optimization
Main articles: semidefinite programming, convex optimization

Examples

Positive polynomials on Euclidean space

A positive polynomial always gives a positive number, no matter what values you choose. For one variable, this happens if the polynomial can be written as the sum of two squares. However, this idea doesn't always work for more than one variable. In 1967, a mathematician named Theodore Motzkin found a special polynomial that is always non-negative but cannot be written as a sum of squares.

Positive polynomials on polytopes

For simpler polynomials with degree one, there is a neat way to check if they stay positive inside certain shapes called polytopes. For more complex shapes and higher-degree polynomials, there are special theorems that help describe these polynomials using combinations of simple building blocks.

Positive polynomials on semialgebraic sets

For more general shapes called semialgebraic sets, there are advanced results that describe how to represent positive polynomials. These results vary depending on the specific type of set.

Positive Hermitian polynomials

When dealing with special kinds of polynomials that use complex numbers and their opposites, there are results that show when these polynomials can be written in a particular form involving sums of squares. This helps in understanding their properties and solving related problems.

Generalizations of positivstellensatz

The idea of a positivstellensatz can be used with other types of math expressions, such as signomials, trigonometric polynomials, polynomial matrices, and special kinds of functions studied in areas like o-minimal structures. These ideas help mathematicians learn more about how positive values work in complicated situations.