Positive polynomial

In mathematics, a positive polynomial is a special kind of math expression that always gives a positive result when we plug in numbers from a certain group or set. Similarly, a non-negative polynomial always gives a result that is zero or higher, never negative. These ideas help mathematicians understand how polynomials behave on different parts of space.

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 no matter which numbers from that set we use, the result will always be greater than zero. This concept is important in many areas of math, helping to solve problems and prove theories.

These polynomials are studied in relation to sets within Euclidean space, which 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 related to this topic, see the Krivine–Stengle Positivstellensatz.

Positivstellensatz and nichtnegativstellensatz

Some special sets 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 simple example of a positive polynomial is one that always gives a positive number no matter what values you put in. 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 showed 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 also be applied to other types of mathematical expressions, such as signomials, trigonometric polynomials, polynomial matrices, and even to special kinds of functions studied in areas like o-minimal structures. These generalizations help mathematicians understand how positive values behave in more complex situations.