Finite Solvability

Definition 4.3.8 (Finite Solvability) LetF ⊆L[x1, . . ., xn] be a sys-

tem of polynomials. F is said to be finitely solvable if: 1. F is solvable.

2. The system of polynomial equations has finitely many zeroes. We will see that this exactly corresponds to the case when the ideal generated by F is a properzero-dimensional ideal. Also, we will see that this corresponds exactly to the case when we can find a set of generators of (F), expressible in a strongly triangular form.

Theorem 4.3.9 Let F _⊂L[x1, . . ., xn] be a system of polynomial equa-

tions. Then the following three statements are equivalent: 1. F is finitely solvable;

2. (F)is a proper zero-dimensional ideal; and

3. If G is a Gr¨obner basis of (F) with respect to >

LEX, then G can be

146 Solving Systems of Polynomial Equations Chapter 4

proof.

(1_⇒2):

AsF is solvable, (F)₆= (1), i.e., (F)_∩L= (0). Assume that

hξ1,1, . . . , ξ1,ni .. .

hξm,1, . . . , ξm,ni are the finite set of common zeros ofF. Define

f(xi) = (xi−ξ1,i)· · ·(xi−ξm,i).

We see that f(xi) is a degreemunivariate polynomial inxi that vanishes at all common zeroes ofF. Thus

∃q >0 hf(xi)q _∈(F)i. Thus,

(F)∩L[xi]6= (0),

and (F) is zero-dimensional. Also, since (0) (F) (1), (F) is also proper.

(2_⇒3):

Since (F) is a proper zero-dimensional ideal, we have (F)_∩L= (0),and

∀xi h (F)_∩L[xi]₆= (0)i, i.e., ∀xi ∃f(xi)∈L[xi] h f(xi)_∈(F)i. SinceGis a Gr¨obner basis of (F), we see that

Hmono(f(xi)) =xDi

i ∈Head(G). Together with the fact that (G_∩L) = 0, we get

∃gi∈G h Hterm(g) =xdi i i , di≤Di. Since we have chosen >

LEX as our admissible ordering,

gi∈L[xi, . . . , xn]\L[xi+ 1, . . . , xn]

and it follows that for alli(0≤i < n) there exists agi+1∈Gi such that gi+1 has a monomial of the form a·xdi+1 (a ∈ L and d > 0). Thus, G,

a Gr¨obner basis of (F) with respect to >

LEX

, can be expressed in strongly triangular form.

(3_⇒1):

LetI= (F) = (G), be the ideal generated byF. It follows that

Z(I) =_Z(F) =_Z(G). Thus it suffices to show thatI is finitely solvable.

1. Since (G_∩L) =I_∩L= (0), 1_6∈I, andI is solvable.

2. We will prove by induction on i that for all i (0 _≤ i < n), the ith

elimination ideal, Ii has finitely many zeroes. We recall that by a previous theorem

Ii = (Gbi)

where Gbi =Snj=iGj, andGbi is in strongly triangular form.

•Base Case: i=n₋1.

Gn−1consists of univariate polynomials inxn. SinceGn−1is strongly

triangular, there is some polynomialp(xn) inGn−1of maximum de-

gree dn. Thus p(xn) has finitely many zeros (at most dn of them). Since we are looking for common zeros of In−1, and since p(xn) ∈ In−1, we see thatIn−1 has finitely many zeros (not more thandn).

•Induction Case: i < n₋1.

By the inductive hypothesis, the (i+ 1)th _{elimination ideal I}

i+1 has

finitely many zeroes, sayDi+2 of them.

Let Π be the projection map defined as follows: Π : An−i→An−i−1

: _hξi+1, ξi+2, . . . , ξni 7→ hξi+2, . . . , ξni. We partition the zero set of the ith _{elimination ideal Ii,}

Z(Ii) into equivalence classes under the following equivalence relation: P, Q_∈

Z(Ii)

P _∼Q iff Π(P) = Π(Q).

By Theorem 4.3.3, and the inductive hypothesis, the number of equiv- alence classes is finite, in fact, less than or equal toDi+2. Letp(xi+1, xi+2, . . ., xn) ∈ Gbi be a polynomial containing a monomial of the form a_·xdi+1

i+1 (a ∈ L and di+1 > 0)—assume that di+1 takes the

highest possible value. If

[P]∼={Q: Π(Q) =hξi+2, . . . , ξni}

is an equivalence class of P, a common zero of Ii, then ξ (where Q=_hξ, ξi+2,. . .,ξn)∈[P]∼) is a zero of the univariate polynomial p(xi+1,ξi+1,. . .,ξn). Thus

|[P]∼| ≤di+1

148 Solving Systems of Polynomial Equations Chapter 4

(-1, +1)

(+1, -1)

Figure 4.1: The zeros ofx1x2+ 1 = 0 andx22−1 = 0.

The above argument also provides an upper bound on the number of zeros of the system of polynomialsF, which is

d1·d2· · ·dn

wherediis the highest degree of a term of the form xdii of a polynomial in Gi−1.

Example 4.3.9 Suppose we want to solve the following system of polyno- mial equations:

{x1x2+ 1, x22−1} ⊆C[x1,x2].

The zeros of the system are (₋1,+1) and (+1,₋1), as can be seen from Figure 4.1.

Clearly the system is finitely solvable.

Now if we compute a Gr¨obner basis of the above system with respect to >

LEX (withx1LEX> x2), then the resulting system is strongly triangular, as

given below:

{x1+x2, x22−1}.

We solve forx2to getx2={+1,−1}. After substituting these values forx2

in the first equation, we get the solutions (x1, x2) ={(−1,+1),(+1,−1)}.

Application: Finite Solvability

FiniteSolvability(F)

Input: F ={f1,. . .,fr} ⊂L[x1,. . .,xn],

L= An algebraically closed field.

(-1,+1)

(+1,-1)

Figure 4.2: The zeros ofx1+x2= 0 andx22−1 = 0.

Compute G, the Gr¨obner basis of (F) with respect to >

LEX

. Output

True, ifGis solvable, and is in strongly triangular form;False, otherwise.