For a given field
and positive integer , the -dimensional
affine space over
is the set
|
(4.51) |
For our purposes in this section, an affine space can be considered as
a vector space (for an exact definition, see [438]). Thus,
is like a vector version of the scalar field
. Familiar
examples of this are
,
, and
.
A polynomial in
can be converted into
a function,
|
(4.52) |
by substituting elements of
for each variable and evaluating the
expression using the field operations. This can be written as
, in which each denotes an element of
that is substituted for the variable .
We now arrive at an interesting question. For a given , what are
the elements of
such that
? We could
also ask the question for some nonzero element, but notice that this
is not necessary because the polynomial may be redefined to formulate
the question using 0. For example, what are the elements of
such that
? This familiar equation for
can be
reformulated to yield: What are the elements of
such that
?
Let
be a field and let
be a set of
polynomials in
. The set
for all |
(4.53) |
is called the (affine) variety defined by
. One interesting fact is that unions and
intersections of varieties are varieties. Therefore, they behave like
the semi-algebraic sets from Section 3.1.2, but for
varieties only equality constraints are allowed. Consider the
varieties
and
. Their
intersection is given by
|
(4.54) |
because each element of
must produce a 0 value for each of
the polynomials in
.
To obtain unions, the polynomials simply need to be multiplied. For
example, consider the varieties
defined as
|
(4.55) |
and
|
(4.56) |
The set
is obtained by forming the
polynomial
. Note that
if either
or
. Therefore,
is a variety. The varieties and were
defined using a single polynomial, but the same idea applies to any
variety. All pairs of the form must appear in the argument
of if there are multiple polynomials.
Steven M LaValle
2020-08-14