Inferring the Topology of Gene Regulatory Networks: An Algebraic Approach to Reverse Engineering - Mathematical Concepts and Methods in Modern Biology - page 85

Since the remaining discussion assumes that the coefficient field has positive char-
acteristic, we can simplify the notation. We will denote the quotient polynomial ring
simply as x n and elements in the ring as polynomials with usual represen-

tation. That is, while an element of the quotient ring is of the form f

F x 1 ,...,

( x 1 ,..., x n ) +

x 1

x n , we will instead write it as f x n

− x 1 ,..., − ( x 1 ,..., x n ) , understanding that

all ring elements have been reduced modulo the ideal x p
1

x n .
Let F j be the set of all polynomials that fit the data for node j as described above.
We solve the problem similarly to solving a system of nonhomogeneous linear equa-
tions in that we construct a “particular” polynomial and the set of “homogeneous”
polynomials. The “particular” polynomial is any that interpolates or fits the given data.
There are numerous methods for constructing an interpolating polynomial function.
Here we use a formula based on the ring-theoretic version of the Chinese Remainder
Theorem:

x n

− x 1 ,..., −

m

f j ( x ) = s i + 1 , j r i ( x ),

i = 1

where r i ( is a polynomial that evaluates to 1 on s i and 0 on any other input, and x

represents x 1 ,

x )

x 2 ,..., x n . Specifically,

m

p − 2

r i ( x ) =

1 (

s i − s k ) ( x − s k ),

k =

where k . For a detailed description

of the construction, see [ 25 ]. For the set of “homogeneous” polynomials, we use the
Ideal-Variety Correspondence from algebraic geometry [ 19 ]. We view the input data
{

is the first coordinate in which s i = s k and i =

as a variety , or a set of roots of a system of polynomials equations. The
Ideal-Variety Correspondence states how to construct the ideal of polynomials that
vanish on the data. To each input state s i ,for1

s 1 ,..., s m }

i m , we associate the (maximal)

ideal

(

s i ) = x 1 − s i 1 ,..., x n − s in . I

(

s i )

Each ideal I is the set of polynomials that evaluate to 0 on the single data point

s i . According to the correspondence, the intersection of these ideals contains all
polynomials that vanish on the union of the input data. So the ideal of “homogeneous”,
or vanishing , polynomials is

m

I ( { s 1 ,..., s m } ) = I ( s i ),

i = 1

which we denote by I for simplicity. Therefore, the set F j is described by f j + I :=

{ f j + h | h ∈ I } .