Davenport, James H.

England, Matthew

English

Cylindrical Algebraic Decomposition (CAD) by projection and lifting requires many iterated univariate resultants. It has been observed that these often factor, but to date this has not been used to optimise implementations of CAD.</p

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Iterated Resultants in CADJames H. Davenport1, Matthew England21University of Bath, Faculty of Science, Department of Computer Science, Bath, UK2Coventry University, Centre for Computational Science and Mathematical Modelling, Coventry, UKAbstractCylindrical Algebraic Decomposition (CAD) by projection and lifting requires many iterated univariateresultants. It has been observed that these often factor, but to date this has not been used to optimiseimplementations of CAD. We continue the investigation into such factorisations, writing in the specificcontext of SC2.KeywordsCylindrical Algebraic Decomposition, Resultant, Gröbner Basis1. IntroductionThe resultant of two polynomials is a polynomial formed of their coefficients that is equal tozero if and only if the two original polynomials have a common root. Resultants are a widelyused tool in symbolic computation, and in satisfiability checking over non-linear arithmetic.In particular, they are a key ingredient of Cylindrical Algebraic Decomposition (CAD) [1]which in its traditional projection and lifting form requires many iterated univariate resultantcalculations.[1, pp. 177–178] suggests that iterated resultants, where there are “common ancestors” tendto factor. This was apparently responded to by van der Waerden in a letter [2], which alas wehave not seen, but the letter’s contents are taken up again in [3]. There are further developmentsin [4, 5]. [3] is based on the theory in [6], which [7] notes has been deleted from more recenteditions (such as [8]). [4] is based on [7]. Despite this factorisation being observed since theinception of CAD, we are not aware of any optimisations in CAD implementations in regardsto it.The purpose of this paper is to look at the connections of results on such factorisations withCylindrical Algebraic Decomposition (CAD) [1] and also Cylindrical Algebraic Coverings (CAC)[9], a recent algorithm that was formed out of the SC2 community via a reworking of CADtheory to better suit the SMT context.For CAD, we assume that we are constructing a CAD for a specific Boolean formula Φ, ratherthan just a set of polynomials. For CAC, we again assume we are looking for SAT/UNSAT for aspecific Boolean formula Φ.8th International Workshop on Satisfiability Checking and Symbolic Computation, July 28, 2023, Tromsø, Norway,Collocated with ISSAC 2023$ masjhd@bath.ac.uk (J. H. Davenport); Matthew.England@coventry.ac.uk (M. England) https://people.bath.ac.uk/masjhd (J. H. Davenport); https://matthewengland.coventry.domains (M. England) 0000-0002-3982-7545 (J. H. Davenport); 0000-0001-5729-3420 (M. England)© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).CEURWorkshopProceedingshttp://ceur-ws.orgISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)2. TheoryWe are grateful to [10] for a clear exposition of the results in [7], which we have borrowed.Definition 1. Given 𝑟 homogeneous polynomials 𝐹1, . . ., 𝐹𝑟 in 𝑥1, . . ., 𝑥𝑛, with indeterminatecoefficients comprising a set 𝐴, an integral polynomial 𝑇 in these indeterminates (that is, 𝑇 ∈ Z[𝐴])is called an inertia form for 𝐹1, . . ., 𝐹𝑟 if 𝑥𝜏𝑖 𝑇 ∈ (𝐹1, . . ., 𝐹𝑟), for suitable 𝑖 and 𝜏 .Van der Waerden observes that the inertia forms comprise an ideal 𝐼 of Z[𝐴], and he showsfurther that 𝐼 is a prime ideal of this ring. It follows from these observations that we may takethe ideal I of inertia forms to be a resultant system for the given 𝐹1, . . ., 𝐹𝑟 in the sense that forspecial values of the coefficients in 𝐾 , the vanishing of all elements of the resultant system isnecessary and sufficient for there to exist a non-trivial solution to the system 𝐹1 = 0, . . ., 𝐹𝑟 = 0in some extension of 𝐾 .Now consider the case in which we have 𝑛 homogeneous polynomials in the same number𝑛 of variables. Let 𝐹1, . . ., 𝐹𝑛 be 𝑛 generic homogeneous forms in 𝑥1, . . ., 𝑥𝑛 of positive totaldegrees 𝑑1, . . ., 𝑑𝑛. That is, every possible coefficient of each 𝐹𝑖 is a distinct indeterminate, andthe set of all such indeterminate coefficients is denoted by 𝐴. Let 𝐼 denote the ideal of inertiaforms for 𝐹1, . . ., 𝐹𝑛. Proofs of the following two propositions may be found in [11].Proposition 1. [10, Proposition 5] 𝐼 is a nonzero principal ideal of Z[𝐴]: 𝐼 = (𝑅), for some𝑅 ̸= 0. 𝑅 is uniquely determined up to sign. We call 𝑅 the (generic multipolynomial) resultant of𝐹1, . . ., 𝐹𝑛.Proposition 2. [10, Proposition 6] The vanishing of 𝑅 for particular 𝐹1, . . ., 𝐹𝑛 with coefficientsin a field 𝐾 is necessary and sufficient for the existence of a non-trivial zero of the system 𝐹1 =0, . . ., 𝐹𝑛 = 0 in some extension of 𝐾 .The above considerations also lead to the notion of a resultant of 𝑛 non-homogeneouspolynomials in 𝑛− 1 variables. For a given non-homogeneous 𝑓(𝑥1, . . ., 𝑥𝑛−1) over 𝐾 of totaldegree d, we may write 𝑓 = 𝐻𝑑+𝐻𝑑−1+ · · ·+𝐻0, where the 𝐻𝑗 are homogeneous of degree 𝑗.Then 𝐻𝑑 is known as the leading form of 𝑓 . Recall that the homogenization 𝐹 (𝑥1, . . ., 𝑥𝑛) of 𝑓is defined by 𝐹 = 𝐻𝑑+𝐻𝑑−1𝑥𝑛+ · · ·+𝐻0𝑥𝑑𝑛𝑛 . Let 𝑓1, . . ., 𝑓𝑛 be particular non-homogeneouspolynomials in 𝑥1, . . ., 𝑥𝑛−1 over 𝐾 of positive total degrees 𝑑𝑖, and with leading forms 𝐻𝑖,𝑑𝑖 .We set res(𝑓1, . . ., 𝑓𝑛) = res(𝐹1, . . ., 𝐹𝑛) , where 𝐹𝑖 is the homogenization of 𝑓𝑖. Then we havethe following (see proof in [11]).Proposition 3. [10, Proposition 7] The vanishing of res(𝑓1, . . ., 𝑓𝑛) is necessary and sufficient foreither the forms 𝐻𝑖,𝑑𝑖 to have a common nontrivial zero over an extension of 𝐾 ,or the polynomials 𝑓𝑖 to have a common zero over an extension of 𝐾 .Observe that the common zeros of the 𝑓𝑖 correspond to the affine solutions of the system, whereasthe nontrivial common zeros of the leading forms correspond to the projective solutions on thehyperplane at infinity.553. Iterated Resultants: An ExampleConsider these polynomials:𝑓 = 𝑦2 + 𝑧2 + 𝑥+ 𝑧 − 1,𝑔 = −𝑥2 + 𝑦2 + 𝑧2 − 1,ℎ = 𝑥2 + 𝑦 + 𝑧.3.1. First variable orderingUnder variable ordering 𝑧 ≻ 𝑦 ≻ 𝑥 we may calculate the iterated resultant:res𝑦(res𝑧(𝑓, 𝑔), res𝑧(𝑓, ℎ)) = 5𝑥8 + 16𝑥7 + 14𝑥6 − 2𝑥5 − 12𝑥4 − 8𝑥3 + 3𝑥2 + 2𝑥= 𝑥(︀5𝑥3 + 6𝑥2 − 3𝑥− 2)︀⏟  ⏞  spurious(︀𝑥2 + 𝑥+ 1)︀ (︀𝑥2 + 𝑥− 1)︀⏟  ⏞  genuine.(1)We define the meaning of the labels below. An alternative computational path may havecalculated similarlyres𝑦(res𝑧(𝑓, 𝑔), res𝑧(𝑔, ℎ)) = 5𝑥8 + 16𝑥7 + 18𝑥6 + 8𝑥5 − 5𝑥4 − 8𝑥3 − 2𝑥2 + 1=(︀𝑥2 + 𝑥+ 1)︀ (︀𝑥2 + 𝑥− 1)︀⏟  ⏞  genuine(︀5𝑥4 + 6𝑥3 + 𝑥2 − 1)︀⏟  ⏞  spurious. (2)The final choice would have been to calculate,res𝑦(res𝑧(𝑓, ℎ), res𝑧(𝑔, ℎ)) = 2𝑥4 + 4𝑥3 + 2𝑥2 − 2= 2(︀𝑥2 + 𝑥+ 1)︀ (︀𝑥2 + 𝑥− 1)︀⏟  ⏞  genuine. (3)Up to constants (3) divides (2) and (1), but this need not happen in general. What does happenin general is that, if we consider a Gröbner Basis,Basisplex(𝑓, 𝑔, ℎ) ={︀𝑥4 + 2𝑥3 + 𝑥2 − 1, 𝑦 − 𝑥, 𝑥2 + 𝑥+ 𝑧}︀, (4)then we see that the basis polynomial in 𝑥 only divides all three iterated resultants and in factis res(𝑓, 𝑔, ℎ) in the sense of §2. In this example, it is also (3), but again this need not happen ingeneral.The labels above are made in regards to the roots of the tagged resultant factors. The roots ofthe part we have labelled as “genuine” are{𝑥 : ∃𝑦∃𝑧𝑓(𝑥, 𝑦, 𝑧) = 𝑔(𝑥, 𝑦, 𝑧) = ℎ(𝑥, 𝑦, 𝑧) = 0}, (5)whereas the roots of the part we have labelled as “spurious” are{𝑥 : ∃𝑦 (∃𝑧1𝑓(𝑥, 𝑦, 𝑧1) = 𝑔(𝑥, 𝑦, 𝑧1) = 0 ∧ ∃𝑧2 ̸= 𝑧1𝑓(𝑥, 𝑦, 𝑧2) = ℎ(𝑥, 𝑦, 𝑧2) = 0)} . (6)They are “spurious” in the sense that they do not go on to form true triple roots. Nevertheless,they are 𝑥 values above which the topology changes, so they cannot always be discarded. Notethat §2 implies that there is always a neat factorisation (over Z if that was the original ring)into “genuine” versus “spurious”.563.2. Second variable orderingWhat happens if we take the variables in a different order? In ordering 𝑥 ≻ 𝑦 ≻ 𝑧 we have:res𝑦(res𝑥(𝑓, 𝑔), res𝑥(𝑓, ℎ)) = (𝑧2 − 1)2, (7)res𝑦(res𝑥(𝑓, 𝑔), res𝑥(𝑔, ℎ)) = (𝑧2 − 1)4, (8)res𝑦(res𝑥(ℎ, 𝑔), res𝑥(𝑓, ℎ)) = (𝑧2 − 1)4, (9)andBasisplex(x,y,z)(𝑓, 𝑔, ℎ) ={︀𝑧2 − 1, 𝑦2 + 𝑦 + 𝑧, 𝑥− 𝑦}︀. (10)I.e. no spurious roots were uncovered with this ordering. The question of CAD variable orderingis well studied and known to greatly effect the complexity of CAD both in practice [12] andtheory [13]. The introduction of spurious factors in some orderings but not others may be asignificant contributing factor to this.4. When Can Spurious Factors be Discarded?This section is not a complete classification on when spurious factors may be discarded, but itis a start.4.1. During CAD with multiple equational constraintsMcCallum [14] introduced the concept of multiple equation constraints, i.e. the case whenΦ ≡ 𝑓1 = 0 ∧ 𝑓2 = 0 ∧ · · · 𝑓𝑘 = 0 ∧ Φ(𝑓𝑘+1, . . . , 𝑓𝑚). (11)Here McCallum projects just res𝑥𝑛(𝑓1, 𝑓𝑖) and disc𝑥𝑛(𝑓𝑖) (as well as various coefficients, whichdo not contribute to the degree explosion).But since 𝑓1 = 0 and 𝑓2 = 0, we know that res𝑥𝑛(𝑓1, 𝑓2) = 0 also. Hence all the res𝑥𝑛(𝑓1, 𝑓𝑖)are equational constraints in 𝑥1, . . . , 𝑥𝑛−1. Thus the next projection isres𝑥𝑛−1(res𝑥𝑛(𝑓1, 𝑓2), res𝑥𝑛(𝑓1, 𝑓𝑖)), (12)res𝑥𝑛−1(res𝑥𝑛(𝑓1, 𝑓2),disc𝑥𝑛(𝑓𝑖)) and numerous discriminants.In this case, we are only interested in the genuine zeros, as away from these the formulawill be uniformly false and thus further refinement is unnecessary. So we can replace (12) byres(𝑓1, 𝑓2, 𝑓𝑖).If the 𝑓𝑖 have degree 𝑑 in each 𝑥𝑖, then the equivalent of (12) after 𝑘 eliminations (i.e.eliminating all equational constraints) has degree 𝑂(︀(2𝑑)𝑑2𝑘)︀(doubly exponential), whereasres(𝑓1, . . . , 𝑓𝑘) has degree 𝑂(︀𝑑𝑘)︀(the Bézout bound). We note that [15] observed that useof 𝑘 equational constraints reduces the double exponent of 𝑚 from 𝑛 to 𝑛 − 𝑘: the presentobservations show that the same reduction applies to the double exponent of 𝑑, at least inasmuchas the nested resultants are concerned.Though it would have to be proved, it seems very likely that the same conclusions wouldapply to equational constraints with the Lazard projection [16]. Here, there are challenges with“curtains” [17], which are the same as the regions of nullification in [18].574.2. During CACIn CAC [9], each polynomial has (at least one) explicit reason for being where it is in thecomputation. For example, res𝑥𝑛(𝑓1, 𝑓2) might be in the computation because of a specific root𝛼, where it is the case for 𝑥𝑛−1 > 𝛼 (until the next point) the regions ruled out by 𝑓1 and 𝑓2overlap, whereas for 𝑥𝑛−1 < 𝛼 we need a further reason to rule out regions. The same mightbe true of res𝑥𝑛(𝑓1, 𝑓3), needed because of a specific root 𝛽. Then (12) tracks where 𝛼 and 𝛽meet. Hence in this context we are interested only in genuine roots, and again we can replace(12) by res(𝑓1, 𝑓2, 𝑓𝑖).We would need to work this through precisely with an implementation of CAC, which hasyet to be done.5. Detecting Spurious FactorsIn the examples above the factors were marked as “spurious” or “genuine” via manual analysisto see if the roots of the factors led to common zeros or not. Are there alternatives to suchmanual detection?We note that in some cases we can discard factors with based on their degree, when thisbreaches the Bézout Bound on the true multivariate resultant. I.e., if res𝑦(res𝑧(𝑓, 𝑔), res𝑧(𝑓, ℎ))has an irreducible factor of degree > 𝑑3, it must be spurious and can be discarded. Since it iscommon for CAD implementation to factor polynomials, this is a cheap, if incomplete, test.Example 1. For example, the following three 3-variable polynomials were created randomly inMaple to have total degree 5:𝑓 = −34𝑥2𝑧3 − 20𝑦5 + 7𝑥2𝑦2 − 43𝑦3𝑧 + 63𝑥+ 16𝑧,𝑔 = 13𝑥𝑧4 − 27𝑧4 − 21𝑥𝑦2 + 30𝑦𝑧 − 42𝑥− 81,ℎ = −65𝑥𝑧4 + 13𝑧5 + 30𝑥3𝑧 + 17𝑥𝑦3 + 25𝑦𝑧 + 78.Then 𝑟𝑒𝑠𝑦(𝑟𝑒𝑠𝑧(𝑓, 𝑔), 𝑟𝑒𝑠𝑧(𝑓, ℎ)) factors into a constant times two irreducible polynomials: oneof degree 378 and the other of degree 89. With no further computation we can identify the firstas spurious since its degree is greater than 53 = 125. The second could be genuine, or be anotherspurious factor: we may check manually that it is indeed genuine.In an example where we have multiple factors below the bound we could work through themin turn keeping count of the sum of degrees of genuine factors as we uncover then, in each casereducing the degree bound accordingly for any further factors to be investigated as genuine.6. ConclusionsThere is much to be done to develop these ideas.1. In §4.1, we have only looked at the resultants, not the discriminants, and indeed only atresultants of resultants. Undoubtedly something similar can be said about, for exampleres𝑦(res𝑧(𝑓, 𝑔), disc𝑧(𝑓)), (13)58but we have not explored this fully yet. We observe that, in the case of the polynomialsfrom Example 1, (13) is a perfect square, and this seems to be true in general. We wouldneed a complete solution for resultants of discriminants, discriminants of resultants anddiscriminants of discriminants in order to need to remove the caveat in italics towardsthe end of §4.1.2. As stated in §4.2, the “genuine parts of resultants” idea would need to be worked throughan implementation of CAC.3. If we look at (3), we see that this polynomial, which is the “genuine” part, factors further,and one factor has no real roots. Hence this factor can be discarded, though there is notmuch benefit, since we are at the univariate phase. Nevertheless, this shows that eventhe “genuine” part may still be overkill for real geometry. Can wea) detect that a factor of a resultant etc. has no real components; andb) use this to further reduce the polynomials? Furthermore,c) can we make any meaningful statement about the complexity implications of this?AcknowledgementsBoth authors are supported by the UK’s EPSRC, via the DEWCAD Project, Pushing Back theDoubly-Exponential Wall of Cylindrical Algebraic Decomposition; grant numbers EP/T015713/1and EP/T015748/1.We are also grateful to Gregory Sankaran and Ali Uncu for many useful conversations.References[1] G. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic De-composition, in: Proceedings 2nd. GI Conference Automata Theory & Formal Lan-guages, volume 33 of Springer Lecture Notes in Computer Science, 1975, pp. 134–183.doi:10.1007/3-540-07407-4_17.[2] B. van der Waerden, About [1], Private communication to G.E. Collins (1975).[3] S. McCallum, Factors of Iterated Resultants and Discriminants, J. Symbolic Comp. 27(1999) 367–385. doi:10.1006/jsco.1998.0257.[4] L. Busé, B. Mourrain, Explicit Factors of Some Iterated Resultants and Discriminants, Math.Comp. 78 (2009) 345–386. doi:10.1090/S0025-5718-08-02111-X.[5] D. Lazard, S. McCallum, Iterated Discriminants, J. Symbolic Comp. 44 (2009) 1176–1193.doi:10.1016/j.jsc.2008.05.006.[6] B. van der Waerden, Modern Algebra Vol. II (trans. F. Blum), Frederick Ungar (1950).[7] J. Jouanolou, Le Formalisme du Résultant, Advances in Mathematics 90 (1991) 117–263.doi:10.1016/0001-8708(91)90031-2.[8] B. van der Waerden, Modern Algebra Vol. II (trans. F. Blum and J.R. Schulenberger),Frederick Ungar (1970).[9] E. Ábrahám, J. Davenport, M. England, G. Kremer, Deciding the Consistency of Non-LinearReal Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic59Coverings, Journal of Logical and Algebraic Methods in Programming 119 (2021), Article100633. doi:10.1016/j.jlamp.2020.100633.[10] S. McCallum, F. Winkler, Differential Resultants, ITM Web of Conferences Article 0100520 (2018). doi:10.1051/itmconf/20182001005.[11] S. McCallum, F. Winkler, Resultants: Algebraic and Differential, Technical Report RISC18-08 Johannes Kepler University, 2018.[12] T. del Rio, M. England., New Heuristic to Choose a Cylindrical Algebraic DecompositionVariable Ordering Motivated by Complexity Analysis, in: F. Boulier, M. England, T.M.Sadykov, and E.V. Vorozhtsov (Ed.) Computer Algebra in Scientific Computing (Proc.CASC 2022), volume 13366 of Springer Lecture Notes in Computer Science, 2022, pp. 300–317.doi:10.1007/978-3-031-14788-3_17.[13] C. Brown, J. Davenport, The Complexity of Quantifier Elimination and Cylindrical Al-gebraic Decomposition, in: C. Brown (Ed.), Proceedings ISSAC 2007, 2007, pp. 54–60.doi:10.1145/1277548.1277557.[14] S. McCallum, On Propagation of Equational Constraints in CAD-Based Quantifier Elimi-nation, in: B. Mourrain (Ed.), Proceedings ISSAC 2001, 2001, pp. 223–230. doi:10.1145/384101.384132.[15] M. England, R. Bradford, J. Davenport, Improving the Use of Equational Constraints inCylindrical Algebraic Decomposition, in: D. Robertz (Ed.), Proceedings ISSAC 2015, 2015,pp. 165–172. doi:10.1145/2755996.2756678.[16] J. Davenport, A. Nair, G. Sankaran, A. Uncu, Lazard-style CAD and Equational Constraints,in: G. Jeronimo (Ed.), Proceedings ISSAC 2023, 2023, pp. 218–226. doi:10.1145/3597066.3597090.[17] A. Nair, Curtains in Cylindrical Algebraic Decomposition, Ph.D. thesis, Uni-versity of Bath, 2021. URL: https://researchportal.bath.ac.uk/en/studentTheses/curtains-in-cylindrical-algebraic-decomposition.[18] S. McCallum, An Improved Projection Operation for Cylindrical Algebraic Decomposition,Ph.D. thesis, University of Wisconsin-Madison Computer Science, 1984. URL: https://www.proquest.com/openview/5a1e6630f4ac77995c62a2fb31f0e9ac/1.60

Iterated Resultants in CAD

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Iterated Resultants in CAD

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