In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a positive polynomial on a particular
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
whose values are positive on that set.
Let ''p'' be a polynomial in ''n'' variables with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s and let ''S'' be a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
ℝ
''n''. We say that:
* ''p'' is positive on ''S'' if ''p''(''x'') > 0 for every ''x'' in ''S''.
* ''p'' is non-negative on ''S'' if ''p''(''x'') ≥ 0 for every ''x'' in ''S''.
* ''p'' is zero on ''S'' if ''p''(''x'') = 0 for every ''x'' in ''S''.
For certain sets ''S'', there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on ''S''. Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
for the most known nullstellensatz.
Examples of positivstellensatz (and nichtnegativstellensatz)
* Globally positive polynomials and
sum of squares decomposition.
** Every real polynomial in one variable and with even
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
is non-negative on ℝ
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
it is a sum of two squares of real ''polynomials'' in one variable. This equivalence does not generalize for polynomial with more than one variable: for instance, the
Motzkin polynomial ''X''
4''Y''
2 + ''X''
2''Y''
4 − 3''X''
2''Y''
2 + 1 is non-negative on ℝ
2 but is not a sum of squares of elements from ℝ
'X'', ''Y''
** A real polynomial in ''n'' variables is non-negative on ℝ
''n'' if and only if it is a sum of squares of real ''rational'' functions in ''n'' variables (see
Hilbert's seventeenth problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original quest ...
and Artin's solution).
** Suppose that ''p'' ∈ ℝ
1, ..., ''X''''n''">'X''1, ..., ''X''''n''is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of even degree. If it is positive on ℝ
''n'' \ , then there exists an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''m'' such that (''X''
12 + ... + ''X''
''n''2)
''m'' ''p'' is a sum of squares of elements from ℝ
1, ..., ''X''''n''">'X''1, ..., ''X''''n''
* Polynomials positive on
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
s.
** For polynomials of degree ≤ 1 we have the following variant of
Farkas lemma Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas.
Farkas' lemma is the key result underpinning the linear programming duality ...
: If ''f'', ''g''
1, ..., ''g''
''k'' have degree ≤ 1 and ''f''(''x'') ≥ 0 for every ''x'' ∈ ℝ
''n'' satisfying ''g''
1(''x'') ≥ 0, ..., ''g''
''k''(''x'') ≥ 0, then there exist non-negative real numbers ''c''
0, ''c''
1, ..., ''c''
''k'' such that ''f'' = ''c''
0 + ''c''
1''g''
1 + ... + ''c''
''k''''g''
''k''.
** Pólya's theorem: If ''p'' ∈ ℝ
1, ..., ''X''''n''">'X''1, ..., ''X''''n''is homogeneous and ''p'' is positive on the set , then there exists an integer ''m'' such that (''x''
1 + ... + ''x''
''n'')
''m'' ''p'' has non-negative coefficients.
** Handelman's theorem: If ''K'' is a compact polytope in Euclidean ''d''-space, defined by linear inequalities ''g''
''i'' ≥ 0, and if ''f'' is a polynomial in ''d'' variables that is positive on ''K'', then ''f'' can be expressed as a linear combination with non-negative coefficients of products of members of .
* Polynomials positive on
semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and inequa ...
s.
** The most general result is
Stengle's Positivstellensatz.
** For compact semialgebraic sets we have
Schmüdgen's positivstellensatz,
Putinar's positivstellensatz and Vasilescu's positivstellensatz. The point here is that no denominators are needed.
** For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.
Generalizations of positivstellensatz
Positivstellensatz also exist for
signomials,
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
s,
polynomial matrices, polynomials in free variables, quantum polynomials, and definable functions on
o-minimal structures.
References
* Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise. ''Real Algebraic Geometry''. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3)
esults in Mathematics and Related Areas (3) 36. Springer-Verlag, Berlin, 1998. x+430 pp. .
* Marshall, Murray. "Positive polynomials and sums of squares". ''Mathematical Surveys and Monographs'', 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. , .
Notes
{{Reflist
See also
*
Polynomial SOS
In mathematics, a form (i.e. a homogeneous polynomial) ''h''(''x'') of degree 2''m'' in the real ''n''-dimensional vector ''x'' is sum of squares of forms (SOS) if and only if there exist forms g_1(x),\ldots,g_k(x) of degree ''m'' such that
h(x) = ...
*
Hilbert's seventeenth problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original quest ...
Real algebraic geometry