In this chapter, we introduced nonlinear algebra and numerical nonlinear algebra by consid-ering the problem of computing the 3264 conics that are tangent to five given conics in the plane. We demonstrated that the 3264 tangent conics for a given instance of five conics correspond to the 3264 isolated solutions of a system of polynomial equations and discussed briefly how we can use numerical nonlinear algebra to quickly compute these solutions. Ad-ditionally, we presented in Proposition 2.1 an instance of five real conics for which there are 3264 real conics tangent to all five given conics. For finding this instance, it was necessary to understand the enumerative geometry approach to deriving the number 3264. The proof of Proposition 2.1 was computer-assisted and relied on the certification of the isolated zeros of a polynomial system. After this first exposure to numerical nonlinear algebra, we will discuss in the next chapter the foundations for the numerical solution of polynomial systems.
3 Background: The Numerical Solution of Polynomial Systems
In this chapter, we outline the foundations for the numerical solution of polynomials systems using homotopy continuation methods. We focus on the necessary concepts and techniques to solve a wide range of systems appearing in applications. For a more comprehensive introduction to the subject, we recommend the book by Sommese and Wampler [SW05], as well as the article by Hauenstein [HS17] for an overview of recent developments.
Before we start, the term “numerical solution” needs more explanation. In general, we want to compute all solutions of a polynomial system and we think of a numerical solution as a point sufficiently close to a solution of the system. For a regular isolated solution, we can make this notion more precise. We require that Newton’s method starting with our numerical solution converges to this solution. If a system has infinitely many solutions, then these can be represented by a finite collection of witness sets (we introduce these in Section 3.7). Each witness set consists of a finite set of isolated solutions and some additional information. Therefore, the numerical solution of polynomial systems can always be reduced to the computation of isolated solutions of polynomial systems.
We focus on the computation of isolated solutions based on the homotopy continuation method. The basic idea for a given polynomial system F that we want to solve is the following.
1. Put the polynomial system F into a family of polynomial systems FQ depending on a parameter set Q. Then there exists a p∈Qsuch that F =Fp ∈ FQ.
2. Solve a general systemFq ∈ FQ.
3. Deform the start systemFq to the target systemFp by moving inside the familyFQ along a path γ : [0,1]→Q with γ(1) =q andγ(0) =p and track the solutions ofFq as it is deformed to Fp.
In the last step, we constructed thehomotopy H(x, t) =Fγ(t)(x)and the problem of tracking a solution is a continuation problem giving the method its name.
From this description, it is not clear why this procedure should allow us to compute all isolated solutions of F. Similarly, there is the question of how to embed F into a family of polynomial systems FQ and how to solve the start system Fq. The system Fq has to be easier to solve otherwise we would end nowhere. We use algebraic geometry to answer these questions in the following sections.
3.1 Preliminaries
We assume basic familiarity with concepts from algebraic geometry and commutative algebra on the level of the undergraduate textbook “Ideals, Varieties, and Algorithms” by Cox, Little, and O’Shea [CLO15]. In the following, we only perform a brief review of the necessary concepts to introduce our notation and refer for details to [CLO15].
In this thesis, a polynomial f is always considered to be an element of the polynomial ringC[x1, . . . , xn] inn variables x1, . . . , xn with coefficients in C. A real polynomial f is a polynomial such thatf and its conjugatef¯are identical. A polynomial systemF :Cn→CN is a collection ofN polynomials f1, . . . , fN ∈C[x1, . . . , xn] with F = (f1, . . . , fN). F is a square system if n = N and overdetermined if N > n. A point x ∈ Cn is a solution (or zero) ofF ifF(x) = 0or equivalently f1(x) =. . .=fN(x) = 0. Algebraically, a polynomial systemF generates an idealI. For an idealI, theaffine variety V(I) is defined by
V(I) ={x∈Cn|f(x) = 0 for all f ∈ I} ⊆Cn.
If the ideal I is generated by F, then we also writeV(F) for the variety defined by I. We refer toV(F) also as thezero set or solution set ofF.
To prove statements related to algebraic varieties, it is immensely helpful to replace Cn with the n-dimensional projective space Pn. Pn is a compact space given by the quotient (Cn+1 \ {0})/∼ where x ∼ y if only if x = λy for some λ ∈ C\ {0}. We write the equivalence class of(x0, x1, . . . , xn)∈Cn+1\ {0}inPnas[x0 :x1:. . .:xn]. A polynomial f ∈ C[x0, x1, . . . , xn] is homogeneous of degree d if for all x ∈Cn+1 and λ∈ C we have f(λx) =λdf(x). Only a homogeneous polynomial is well defined as a map Pn→P.
LetF¯ = (f¯1, . . . , f¯ )N be a system ofN homogeneous polynomials inn+ 1variables. We call such a systemhomogeneous. A homogeneous systemF¯ generates a homogeneous ideal J. For a homogeneous ideal J, theprojective variety V(J)is defined by
V(J) ={x∈Pn|f(x) = 0 for all f ∈ J } ⊆Pn.
As in the affine case, we write V(F¯ )for the projective variety defined by the homogeneous ideal generated byF¯. A homogeneous ideal J also defines an affine variety
V(J) ={x∈Cn+1|f(x) = 0 for all f ∈ J } ⊆Cn+1 called theaffine cone overV(J).
Consider the open set Ui = Pn\ V(xi). Since any point in projective space has some nonzero coordinate, the Ui cover Pn and every point in Ui has a unique representative of
the form (︃
are the standard affine charts of Pnand give Pn the structure of a manifold. It follows that locally every projective variety looks like an affine variety. The construction also generalizes to any nonzero homogeneous linear polynomial ℓ(x) =a0x0+. . .+anxn since every point in Uℓ = Pn\ V(ℓ) has a unique affine representative of the form (x0/ℓ(x), . . . , xn/ℓ(x)) . Thus, we can identify Uℓ with the affine space V(ℓ−1)∼=Cn.
Given a projective variety X=V(F¯ )⊆Pn, the intersection of the affine cone of X with the hyperplane V(xi−1), considered as a subset of Cn, is thedehomogenizationof X with respect to xi. The dehomogenization of X is cut out by F¯ (x1, . . . , xi−1,1, xi+1, . . . , xn). Conversely, assume we have an affine variety X = V(F) ⊆ Cn. We define the projective closure X¯ ofX by introducing a new variablex0 and
X={[1 :x]∈Pn|x∈X}
where the closure is taken in the Zariski topology. To determine defining equations for X¯ is in general a difficult task. The naive strategy is to use thehomogenizationF¯ = (f¯
1, . . . , f¯ is necessary to compute a homogenization of the ideal generated by F. This is in general not the ideal generated by the homogenization of F.
Given an projective varietyX⊂Pnof dimensionk, thedegree ofX, denoted bydeg(X), is the number of intersection points with a general linear space L ⊂Pn of codimension k, i.e.,deg(X) =|X∩L|. The analogous definition holds for affine varieties X⊂Cn. solution if the Jacobian JF(x) ofF atx has full column rank.