Certification of ApproximateRoots of Exact PolynomialSystems

At the 2016 AMS Fall Southeastern Sectional Meeting I will talk about my joint work with Jonathan Hauenstein and ourstudent Tulay Ayyildiz Akoglu on the certification of solutions of polynomial systems.

Agnes Szanto

In the past fifty years, since the development of Buch-berger’s algorithm for computing Gröbner bases,symbolic computation has seen an explosion indevelopment and applications for solving systemsof polynomial equations. More recently, the com-

bination of numerical analysis and algebraic geometry,yielding the field of numerical algebraic geometry, hasproduced new approaches and extended the limits of solv-ing systems of polynomial equations. In general, thesenumerical methods are parallelizable and are produc-ing solutions to large-scale systems. For example, theBertini software package, a general-purpose numerical

Agnes Szanto is associate professor of mathematics at North Car-olina State University. Her email address is aszanto@ncsu.edu.

Agnes was partially supported for this research by NSF GrantCCF-1217557.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1440

solver for polynomial systems, has recently been used tosolve a system of seventy quadratics in seventy variables[4]. An overview of numerical algebraic geometry with afocus on using the software Bertini can be found in [2].

Sophisticatednumerical algebraic geometry techniquesare also able to solve problems that were once thought tobe purely symbolic in nature, corresponding to noncon-tinuous properties of the input polynomial system. Someof the surprising problems that can be handled using nu-merical methods include finding primary decompositions,the genus of irreducible curves, or intersection numbersof Chern classes. However, many of the results producedby numerical algebraic geometry are not certified, as theyare generated using heuristic methods that relax noncon-tinuous properties into continuous ones. The aim of ourwork is to give certification techniques for these noncon-tinuous problems and to demonstrate that certificatescan be computed with not too much extra work, givennumerical data.

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As mentioned above, typically the output of numericalalgebraic geometry computations are not certified, withone notable exception being the special case of nonsingu-lar roots of well-constrained systems. Before explainingour contributions, let me describe𝛼-theory, one of the cer-tification techniques from the extended body of literaturefor this special case.

Let 𝑓 ∶ ℂ𝑛 → ℂ𝑛 be a system of analytic functions andz ∈ ℂ𝑛. Introduced by Smale in 1986,𝛼-theory guaranteesquadratic convergence of the Newton iteration from zto a fixed point if an associated invariant 𝛼(𝑓,z) issmaller than a universal constant. Thus, in the case of anonsingular root of a well-constrained system, 𝛼-theorygives a certificate that z is an approximate zero of 𝑓,i.e., that Newton’s iteration starting from z quadraticallyconverges to a root 𝜉 ∈ ℂ𝑛 of 𝑓. Moreover, 𝛼-theoryalso gives a bound on the distance of z from 𝜉 andcertifies the uniqueness of the root within that distance.This last property allows 𝛼-theory to certify that z ∈ ℝ𝑛

approximates a real root of a real polynomial system,which is used in many applications.

The invariant 𝛼(𝑓,z) depends on all derivativesof 𝑓 evaluated at z. If 𝑓 is a polynomial sys-tem, an upper bound for 𝛼(𝑓,z) can be efficientlyevaluated, which is implemented in the softwarealphaCertified [5]. A fascinating application is acomputerized proof of a conjecture by John EdensorLittlewood, who asked the following question:

Is it possible in 3-space for seven infinite circularcylinders of unit radius each to touch all the oth-ers? Seven is the number suggested by constants.

Bozóki, Lee, and Rónyai [3] proved this conjecture bysetting up a well-constrained polynomial system overℤ with real roots corresponding to solutions of theLittlewood conjecture. They approximated the roots usinga numerical homotopy continuation method and certifiedthat some roots are real using alphaCertified. A modelof a cylinder arrangement solving Littlewood’s conjecturethat they found is shown in Figure 1.

Figure 1. Bozóki, Lee, and Rónyai proved that sevencylinders can all touch each other.

Motivated by such applications, we turned our atten-tion to polynomial systems that are overdetermined orhave singular roots. Note that consistency of an over-determined polynomial system is not a continuous prop-erty of the input, and similarly the multiplicity structureof a given root is destroyed by perturbations of the inputsystem. Existing approaches to certify solutions to suchsystems permit small perturbations of the input system,transforming the problem into a continuous one, allowingthe use of 𝛼-theory or interval arithmetic or other con-tinuous certification methods. However, for applicationssuch as the computerized mathematical proof above, this

is not sufficient: the polynomial system is given exactlywith coefficients from ℤ or ℚ, and we need to certifyapproximate zeroes of the given exact system withoutany perturbation.

Certifying the above noncontinuous properties posedcompletely new challenges and needed new ideas. Letme demonstrate the difficulties in a very simple example:Consider the overdetermined system 𝑓 ∶ ℂ → ℂ2 definedfor any 𝑎 ∈ ℂ by

𝑓(𝑥) = ( 𝑥𝑥2 +𝑎 ) .

Dedieu and Shub developed an 𝛼-theory for overdeter-mined systemsusing the so-calledGauss-Newton iteration,but this only certifies fixed points which may not be roots;e.g., 𝑥 = 0 is a fixed point of the Gauss-Newton iterationbut not a root of 𝑓, unless 𝑎 = 0. In particular, rootsand local minima of ‖𝑓‖ are indistinguishable when usingcontinuous certification methods.

To overcome this problem, Hauenstein and Sottileconsideredusinguniversal lowerbounds for theminimumof positive polynomials over ℤ on a disk. They concludedthat all known bounds were “too small to be practical.”For example, the overdetermined system

𝑓1 ∶= 𝑥1 −12, 𝑓2 ∶= 𝑥2 − 𝑥2

1,… , 𝑓𝑛 ∶= 𝑥𝑛 − 𝑥2𝑛−1, 𝑓𝑛+1 ∶= 𝑥𝑛

has no common roots, but the value of 𝑓𝑛+1 on thecommon root of 𝑓1,… , 𝑓𝑛 is doubly exponentially smallin 𝑛. Since universal lower bounds have to cover theseartificial cases, we cannot expect much improvementusing universal bounds.

This led to the idea of constructing some individualizedwitness for our input polynomial system 𝑓 instead of usinguniversal lower bounds that are often very pessimistic.In our approach we use the so-called Rational UnivariateRepresentation (RUR) as our witness, and we reconstructthis exact representation from approximate data. Theadvantage of our approach, as we shall see, is the abilityto terminate early in cases when the witness for our inputinstance is small.

The main idea of our symbolic-numeric method tocertify overdetermined polynomial systems over ℚ isas follows: Consider 𝑓 = (𝑓1,… , 𝑓𝑚) ∈ ℚ[𝑥1,… , 𝑥𝑛]𝑚 forsome𝑚 > 𝑛, and assume for now that all roots are isolatedand nonsingular. Under these assumptions, the RUR forthe common roots of 𝑓 exists, and the polynomials in theRUR also have rational coefficients. Thus we can computethem exactly, unlike the possibly irrational coordinatesof the common roots of 𝑓. With the exact RUR, whichis a well-constrained system of polynomials, we canuse alphaCertified to certify that a given point is anapproximate root for the RUR and thus for our originalsystem 𝑓.

In principle, one can compute such an RURusing purelyalgebraic techniques, for example, by solving large linearsystems corresponding to resultant or subresultant ma-trices. However, this purely symbolic method would againlead to running times close to the worst case complexitybounds. Instead, we propose a hybrid symbolic-numeric

November 2016 Notices of the AMS 1161

approach using interpolation on the approximate rootsof 𝑓, rational number reconstruction, as well as exactunivariate polynomial remaindering over ℚ. Our hopeis that our method will make the certification of rootsof overdetermined systems practical for cases when theactual size of the RUR is significantly smaller than in theworst case.

The details of this hybrid symbolic-numeric methodare described in our paper [1], which also contains de-tailed complexity analysis, demonstrating that in the caseof successful certification the complexity is polynomialin the input plus the output size. This is a significant im-provement over purely symbolic methods, which alwayshave comparable complexity to our worst-case scenariowhen the certification fails. In other words, our methodreturns certification in many instances very quickly, butin some cases it terminates with failure.

Once we successfully certified nonsingular roots ofoverdetermined systems overℚ, we considered certifyingisolated singular roots of rational polynomial systems.Due to the behavior of Newton’s method near singularroots, standard techniques in 𝛼-theory cannot be appliedto certify such roots even if the polynomial system is wellconstrained. The key tool to handle such multiple rootsis called deflation. Deflation techniques “regularize” thesystem, thereby creating a new polynomial system whichhas a simple root corresponding to the multiple root ofthe original system. In our work, we focus on using adeterminantal form of the isosingular deflation, in whichone simply adds new polynomials to the original systemwithout introducing new variables. The new polynomialsare certain subdeterminants of the Jacobian matrix of thepolynomial system, constructed using exact informationthat one can obtain from a numerical approximation ofthe multiple root. In particular, if the original systemhad rational coefficients, the new polynomials whichremove the multiplicity also have rational coefficients.Thus, this technique provides a reduction to the case ofan overdetermined system over ℚ in the original set ofvariables that has a simple root.

We demonstrate our method on a common benchmarksystem, the Caprasse system, which has 24 regular rootsand 8 roots of multiplicity 4, namely,

𝑔 =

⎡⎢⎢⎢⎢⎢⎢⎣

𝑥31𝑥3 −4𝑥21𝑥2𝑥4 − 4𝑥1𝑥22𝑥3 −2𝑥32𝑥4 −4𝑥21 − 4𝑥1𝑥3 + 10𝑥22 +10𝑥2𝑥4 − 2𝑥1𝑥33 −4𝑥1𝑥3𝑥24 − 4𝑥2𝑥23𝑥4 −2𝑥2𝑥34 −4𝑥1𝑥3 +10𝑥2𝑥4 − 4𝑥23 + 10𝑥24 − 2

2𝑥1𝑥2𝑥4 + 𝑥22𝑥3 −2𝑥1 − 𝑥3𝑥1𝑥24 + 2𝑥2𝑥3𝑥4 − 𝑥1 − 2𝑥3

⎤⎥⎥⎥⎥⎥⎥⎦

.

Since the system is well constrained, numerical approx-imations for the 24 regular roots can be certified usingstandard 𝛼-theory. Here, we consider certifying the mul-tiple roots. When evaluated at each of these multipleroots, the 4× 4 Jacobian matrix 𝐽𝑔=[𝜕𝑔𝑖

𝜕𝑥𝑗]4

𝑖,𝑗=1has rank 2,

with the lower right 2 × 2 block having full rank, whichproperty can be obtained using approximate roots andnumerical rank computations. Thus, we construct the newpolynomial system 𝑓 by appending 𝑔 with the four 3 × 3

minors of 𝐽𝑔 containing the lower right block. Now 𝑓 isan overdetermined polynomial system over ℚ that com-pletely deflates all 8 roots; i.e., they become nonsingularroots of 𝑓.

From the numerical approximations of the eight pointsthat we computed using Bertini, we see that 𝑇 =𝑥1−𝑥2+3𝑥3−3𝑥4 separates the roots. Using interpolationon thenumerical approximationsof the roots correct to sixdigits we constructed the following exact RUR, which wecertified to be correct using exact univariate polynomialremaindering:𝑞 = (𝑇2 + 4/3)(𝑇2 + 12)(𝑇2 −16𝑇+ 76)(𝑇2 + 16𝑇+ 76),𝑟1 = (224/3)𝑇6 − (15232/3)𝑇4 − 6656𝑇2 +505856/3,𝑟2 = −(80/3)𝑇6 − 1856𝑇4 − 41216𝑇2 + 97280/3,𝑟3 = (160/3)𝑇6 −2944𝑇4 − (363008/3)𝑇2 − 972800/3,𝑟4 = (80/3)𝑇6 + 1856𝑇4 + 41216𝑇2 − 97280/3.

This RUR corresponds to a well-constrained system𝑞,𝑞′𝑥1𝑟1,… , 𝑞′𝑥4 −𝑟4) ∈ 𝑄[𝑇, 𝑥1, 𝑥2, 𝑥3, 𝑥4]5 defining the8 roots, now with multiplicity one. Thus we can usealphaCertified to certify that a given point is an ap-proximate root for the RUR and thus for our originalsystems 𝑓 and 𝑔.

References[1] T. A. Akoglu, J. D. Hauenstein, and A. Szanto, Certify-

ing solutions to overdetermined and singular polynomialsystems over ℚ, arXiv:1408.2721, 2014.

[2] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W.Wampler, Numerically solving polynomial systems withBertini, volume 25 of Software, Environments, and Tools,Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, 2013. MR 3155500

[3] S. Bozóki, T.-L. Lee, and L. Rónyai, Seven mutually touch-ing infinite cylinders. Comput. Geom. 48 (2015), no. 2,87–93. MR 3260249

[4] J. Hauenstein and J. McCarthy, Biologically-inspiredlinkage design: computing form from function. SIAM News48 (2015), no. 8.

[5] J. D. Hauenstein and F. Sottile, Algorithm 921: alphaCertified: certifying solutions to polynomial systems, ACMTrans. Math. Software 38 (2012), no. 4, Art. ID 28, 20 pp.MR 2972672

CreditsFigure 1, courtesy of Peter Gal.Photo of Agnes Szanto, courtesy of Scott Hellmann.

ABOUT THE AUTHOR

Research interests of Agnes Szanto in-clude symbolic-numeric computation, computational algebraic geometry, and differential algebra.

Agnes Szanto

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