So far, we discussed numerical methods to compute (all) isolated zeros of a polynomial system F. Since we use numerical methods, we do not obtain exact zeros of F but rather numerical approximations. For many applications, this is sufficient. But for some applica-tions, in particular in pure mathematics, we want tocertify that the obtained approximations correspond to actual zeros of F. Additionally, we want to certify that we found a certain number of distinct zeros, establishing alower bound on the number of solutions of F.
It is of great interest to also certify that we foundallisolated solutions ofF. Unfortunately, this is so far only possible if an upper bound is already established and the lower bound obtained from the certification routine matches the upper bound. Establishing an upper bound for a polynomial system based on a computational method that does not involve Gröbner basis computations is an important open problem. A numerical method to test whether all solutions are found is given by the trace test. However, the trace test does not produce a rigorous certificate. The result of the trace test can be interpreted similarly to the numerical computation of the smallest eigenvalue of a matrix. If the computed smallest eigenvalue is on the order of the machine precision, then you probably conclude that the matrix is singular. But this isnot a proof (or certificate) that the matrix is singular.
3.6.1 Certification
We focus on two strategies to certify solutions to square polynomial systems. Smale’s α -theory [Sma86] and Krawczyk’s method [Moo77]. The restriction to square polynomial systems is necessary since to certify solutions to overdetermined systems additional global information is necessary. In [DHS20], the authors develop various techniques to certify overdetermined systems requiring different global information. A certification technique based on Krawczyk’s method is presented in Chapter 5. There, we also demonstrate an implementation of Krawczyk’s method in HomotopyContinuation.jl that provides a sig-nificant computational improvement over the existing certification methods based on Smale’s α-theory. However, since Smale’sα-theory only requires data from one point, it has valuable features for the theory of computation. In particular, it is the building block for the complex-ity analysis of polynomial homotopy continuation methods [BP09, BC11, BL13, Lai17] and certified path tracking algorithms [BL13].
In [Sma86], Smale introduced the notion of an approximate zero, theα-number and the
α-theorem. For a square polynomial systemF in nvariable, consider the Newton iteration JF(x(j))∆x(j) =F(x(j))
x(j+1) =x(j)−∆x(j) , j= 0,1,2, . . .
starting at the initial guess x(0) ∈Cn where JF is the Jacobian of F. An approximate zero of F is any pointx(0) ∈Cn such that Newton’s method converges quadratically towards a zero of F. This means that the number of correct significant digits roughly doubles with each iteration of Newton’s method.
Definition 3.11 (Approximate zero). The point x(0) ∈ Cn is an approximate zero of F if the Newton iterates x(j) are defined for j= 1,2, . . . and satisfy
∥∆x(j)∥ ≤ (︃1
2 )︃2j−1
∥∆x(0)∥.
If x(0) is an approximate zero, then the true zero x∗ ∈ Cn of F to that the iterates are converging is theassociated zero of x(0).
Smale’s α-theorem gives a sufficient condition for x(0) to be an approximate zero. The theorem uses
γ(F, x) = sup
k≥2
⃦
⃦ 1
k!JF(x)−1DkF(x)⃦⃦
1
k−1 andβ(F, x) = ∥JF(x)−1F(x)∥ (3.10) whereDkF is the tensor of order-k derivatives of F and the tensor JF−1DkF is understood as a multilinear map A: (Cn)k→Cn.
Theorem 3.12 (Smale’s α-theorem [Sma86]). There is a naturally defined number α0 ap-proximately equal to 0.1307 such that if α(F, x(0)) :=β(F, x(0))γ(F, x(0))< α0, thenx(0) is an approximate zero ofF.
To avoid the computation of the γ-number, Shub and Smale [SS93] derived an upper bound for γ(F, x) that can be computed exactly and efficiently. Hence, one can decide algorithmically whetherxis an approximate zero using only the data of the point xitself and F. Hauenstein and Sottile implemented these ideas in the softwarealphaCertified[HS12].
3.6.2 Trace Test
The trace test was first introduced in [SVW02] by Sommese, Verschelde, and Wampler and recently generalized in [LRS18] by Leykin, Rodriguez, and Sottile to subvarieties of products of projective spaces. Suppose we have an irreducible one-dimensional variety X ⊆Cn. The intersection of X with a general hyperplane L consists of deg(X)-many points. The trace of X with respect to the hyperplane L is the coordinate-wise sum of the points in L∩X. Let l(x) be an affine linear polynomial defining L. Denote by Lt the zero set of l(x) +t.
Lt is a hyperplane that depends linearly ontand the trace ofX with respect to Ltdepends ont. We have the following result.
Proposition 3.13. [SVW02, LRS18] Using the notation and definitions above, the trace of X with respect to Lt is affine linear in t. Moreover, the coordinate-wise sum of any non-empty proper subset ofX∩Lt is not affine linear in t.
This leads to the idea of the trace test. Given a subset W ⊆X∩L, denote by Wt the points obtained by trackingW fromX∩L0 toX∩Lt. By construction we haveW =W0. Also, denote by w(t) the sum of the points of Wt. Following Proposition 3.13, w(t) is an affine linear function if and only ifWt=X∩Lt. SinceWtis the result of trackingW along X∩Lt, it follows thatw(t) is an affine linear function if and only ifW =X∩L. The trace test computes for a generalτ ∈C\ {0}the pointsw(0),w(τ)andw(−τ)and tests whether these three points are colinear. By our previous argument, this is with probability one only the case ifW =X∩L. A numerical stable method to test this is to compute the singular valuesσ1≥σ2 ≥σ3 of the matrix
[︄w(0) w(τ) w(−τ)
1 1 1
]︄
.
Here, a row of ones is appended to account for the case that X is a plane curve. The smallest singular valueσ3 is zero if w(t) is affine linear. Thus, a good numerical colinearity test is to check that for the computed singular values the value ofσ3/σ1 is on the order of the machine precision.
Above we assumed thatX⊆Cnis a one-dimensional affine variety. But this is not nec-essary and the trace test can be extended to positive-dimensional irreducible affine varieties by replacing Lt with a pencil of affine linear spaces Mt such that codim(M0) = dim(X). A pencil of affine linear spaces is a family Mt, t∈ C, of affine linear spaces that depends affinely on the parameter t. Mt is the span of an affine linear space H and a pointt on a line h that is disjoint fromH.
So far, the presented trace test does not apply to isolated solutions of a general param-eterized polynomial systemF(x;p). For this, we consider again for some irreducible variety Q⊂Cm the incidence variety
Z ={(x, p)∈Cn×Q|F(x;p) = 0} ⊆Cn×Q .
and the projectionπ:Z →Q,(x, p)↦→ponto the second factor. We consider an irreducible subvariety Y ⊆ Z with the property that there exists an open set U ⊆ Q such that for all q ∈ U the fiber π|−1Y (q) is finite and has the same cardinality d. In Section 3.5, we demonstrated how the monodromy method allows us to compute all elements ofπ|−1Y (q)for q∈U. But we had the difficulty that we needed a heuristic to stop the computation.
IfU is non-empty, then the varietyY ⊆Cn×Q is anm-dimensional irreducible variety.
Consider the closureY ofY in the productPn×Pm of projective spacesY. In [LRS18], the authors give a trace test for subvarieties of products of projective spaces. Dehomogenizing the procedure allows us to give a trace test forY. We follow the approach outlined in [MdCR17].
In a first step, we intersect Y with a linear space Λ of codimension m−1 defined by m−1 general affine linear polynomials G in p resulting in the one-dimensional subvariety Y ∩Λ⊆Y ⊆Cn×Cm. This dimension reduction is an application of a version of Bertini’s
theorem [LRS18, Theorem 12] and preserves the irreducibility of Y. Whereas we previously intersected with an affine linear space, we now intersect with an affine bilinear space B defined by the product of two affine linear polynomials ℓ1(x) andℓ2(x) in the unknowns x and p respectively. Since we reduced to a curve in Cn+m ∼=Cn×Cm, we can apply again Proposition 3.13. That is, if we define the family Bt of bilinear spaces by ℓ1(x)ℓ2(p) +t, then the trace of Y ∩Λ with respect to Bt is affine linear in tfor the x andp coordinates.
Moreover, the coordinate-wise sum for any proper non-empty subset of Y ∩Λ∩ Bt is not affine linear int.
Now assume we attempted to compute with the monodromy method all elements of π|−1Y (q) for some q ∈ U and obtained as a result the subset W ⊆ π|−1Y (q). To check for the completeness of W, we want to apply the trace test. For this, we need to choose G(p) andℓ2(p) such that q is the unique solution of G(p) =ℓ2(p) = 0. After choosing a general affine linear polynomial ℓ1(x), we need to start a second monodromy computation for the polynomial system solutions for the monodromy computation. At some point the monodromy computation recoversY ∩Λ∩ B0⊆Cn×Cm. This can be verified by testing that the trace ofY ∩Λwith respect to Bt is colinear. Finally, if the trace is colinear and W × {q} =Y ∩(Cn× {q}), then W =π|−1Y (q) and the first monodromy computation found all solutions.