In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of
Bézout's theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the de ...
, which counts the number of isolated common zeros of a set of
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...
s. This generalization is due to
Igor Shafarevich.
Motivation
Given a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
or a
system of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some Field (mathematics), field .
A ''solution'' of a polynomial system is a se ...
it is often useful to compute or to bound the number of solutions without computing explicitly the solutions.
In the case of a single equation, this problem is solved by the
fundamental theorem of algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
, which asserts that the number of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
solutions is bounded by the
degree of the polynomial, with equality, if the solutions are counted with their
multiplicities
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multipl ...
.
In the case of a system of polynomial equations in unknowns, the problem is solved by
Bézout's theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the de ...
, which asserts that, if the number of complex solutions is finite, their number is bounded by the product of the degrees of the polynomials. Moreover, if the number of solutions
at infinity is also finite, then the product of the degrees equals the number of solutions counted with multiplicities and including the solutions at infinity.
However, it is rather common that the number of solutions at infinity is infinite. In this case, the product of the degrees of the polynomials may be much larger than the number of roots, and better bounds are useful.
Multi-homogeneous Bézout theorem provides such a better bound when the unknowns may be split into several subsets such that the degree of each polynomial in each subset is lower than the total degree of the polynomial. For example, let
be polynomials of degree two which are of degree one in indeterminate
and also of degree one in
(that is the polynomials are ''bilinear''. In this case, Bézout's theorem bounds the number of solutions by
:
while the multi-homogeneous Bézout theorem gives the bound (using
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
)
:
Statement
A multi-homogeneous polynomial is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
that is
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
with respect to several sets of variables.
More precisely, consider positive integers
, and, for , the
indeterminates A polynomial in all these indeterminates is multi-homogeneous of multi-degree
if it is homogeneous of degree
in
A multi-projective variety is a
projective subvariety of the product of
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s
:
where
denote the projective space of dimension . A multi-projective variety may be defined as the set of the common nontrivial zeros of an ideal of multi-homogeneous polynomials, where "nontrivial" means that
are not simultaneously 0, for each .
Bézout's theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the de ...
asserts that homogeneous polynomials of degree
in indeterminates define either an
algebraic set
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of positive
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
, or a zero-dimensional algebraic set consisting of
points counted with their multiplicities.
For stating the generalization of Bézout's theorem, it is convenient to introduce new indeterminates
and to represent the multi-degree
by the linear form
In the following, "multi-degree" will refer to this linear form rather than to the sequence of degrees.
Setting
the multi-homogeneous Bézout theorem is the following.
''With above notation,'' ''multi-homogeneous polynomials of multi-degrees''
''define either a multi-projective algebraic set of positive dimension, or a zero-dimensional algebraic set consisting of'' ''points, counted with multiplicities, where'' ''is the coefficient of''
:
''in the product of linear forms''
:
Non-homogeneous case
The multi-homogeneous Bézout bound on the number of solutions may be used for non-homogeneous systems of equations, when the polynomials may be (multi)-
homogenized without increasing the total degree. However, in this case, the bound may be not sharp, if there are solutions "at infinity".
Without insight on the problem that is studied, it may be difficult to group the variables for a "good" multi-homogenization. Fortunately, there are many problems where such a grouping results directly from the problem that is modeled. For example, in
mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
, equations are generally homogeneous or almost homogeneous in the lengths and in the masses.
References
{{algebraic-geometry-stub
Theorems about polynomials
Algebraic geometry