Probabilistic Saturations and Alt's Problem
Jonathan D. Hauenstein, Martin Helmer
TL;DR
This paper develops probabilistic symbolic and numeric methods using saturations to estimate upper bounds for solutions to Alt's problem, a classical problem in kinematics, with applications to algebraic geometry.
Contribution
It introduces effective probabilistic saturation techniques for counting solutions to polynomial systems, providing bounds and methods applicable beyond Alt's problem.
Findings
Derived bounds on the number of solutions outside the base locus.
Developed methods using finite fields and floating-point computations.
Applicable to problems in algebraic geometry like Newton-Okounkov bodies and intersection invariants.
Abstract
Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods…
Click any figure to enlarge with its caption.
Figure 1| Prime | Successes in 10,000 Trials |
|---|---|
| 8,202 | |
| 9,057 | |
| 9,592 | |
| 9,941 | |
| 9,986 | |
| 9,994 | |
| 9,998 | |
| 9,999 | |
| 10,000 | |
| 10,000 | |
| 10,000 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |
|---|---|---|---|---|---|---|---|---|
| degree | 7 | 43 | 234 | 1108 | 3832 | 8716 | 10858 | 8652 |
| 5 | 4 | 3 | 2 | 1 | 0 | |
|---|---|---|---|---|---|---|
| 176 | 849 | 3352 | 6192 | 8756 | 8492 | |
| 0.16s | 4.66s | 2.50m | 0.52h | 4.69h | 0.84d | |
| 189 | 975 | 3594 | 6803 | 10693 | 8586 | |
| 0.23s | 6.99s | 3.36m | 0.79h | 7.01h | 1.20d | |
| 233 | 1003 | 3566 | 7673 | 10651 | 8644 | |
| 0.27s | 6.26s | 3.94m | 1.24h | 7.44h | 3.45d | |
| 234 | 1059 | 3766 | 8635 | 10739 | 8646 | |
| 0.23s | 7.94s | 4.46m | 1.20h | 7.99h | 1.82d | |
| 234 | 1100 | 3812 | 8716 | 10858 | 8652 | |
| 0.29s | 8.49s | 4.38m | 1.42h | 8.55h | 2.69d | |
| 234 | 1107 | 3832 | 8716 | 10858 | 8652 | |
| 0.34s | 7.53s | 4.64m | 1.41h | 9.68h | 3.11d | |
| 234 | 1108 | 3832 | 8716 | 10858 | 8652 | |
| 0.23s | 8.45s | 4.67m | 1.41h | 8.52h | 2.54d | |
| 234 | 1108 | 3832 | 8716 | 10858 | 8652 | |
| 0.24s | 7.85s | 4.83m | 1.47h | 9.36h | 3.03d | |
| 234 | 1108 | 3832 | 8716 | 10858 | 8652 | |
| 0.24s | 7.96s | 4.83m | 1.56h | 9.11h | 2.91d | |
| 234 | 1108 | 3832 | 8716 | 10858 | ||
| 0.81s | 87.63s | 95.23m | 36.76h | 250.31h | ||
| 234 | 1108 | 3832 | ||||
| 3.08s | 395.85s | 1385.45m |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 7 | 1 | 3 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
| 6 | 1 | 4 | 10 | 20 | 35 | 43 | 43 | 43 | 43 | 43 | 43 |
| 5 | 1 | 5 | 15 | 35 | 70 | 126 | 209 | 234 | 234 | 234 | 234 |
| 4 | 1 | 6 | 21 | 56 | 126 | 252 | 460 | 777 | 1108 | 1108 | 1108 |
| 3 | 1 | 7 | 28 | 84 | 210 | 462 | 920 | 1686 | 2876 | 3832 | 3832 |
| 2 | 1 | 8 | 36 | 120 | 330 | 791 | 1704 | 3362 | 6154 | 8716 | 8716 |
| 1 | 1 | 9 | 45 | 165 | 495 | 1285 | 2981 | 6307 | 10858 | 10858 | 10858 |
| 0 | 1 | 10 | 55 | 220 | 715 | 1999 | 4971 | 8652 | 8652 | 8652 | 8652 |
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Probabilistic Saturations and Alt’s Problem
Jonathan D. Hauenstein
and
Martin Helmer
Abstract.
Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.
1. Introduction
In 1923, Alt [3] realized that finitely many four-bar planar linkages (see Figure 1) can be constructed whose coupler curve interpolates nine general points in the plane. Since he could not determine the exact number himself, he left this as an open problem which was solved numerically using homotopy continuation nearly 70 years later by Wampler, Morgan, and Sommese [40]. In particular, they showed that there are 1442 distinct four-bar coupler curves which pass through nine general points which, together with Roberts cognates [35], yields 4326 distinct four-bar linkages. Although this computation has been repeatedly confirmed using various homotopy continuation methods, e.g., [5, 12, 21, 22, 34, 38], these numerical computations do not preclude the existence of additional solutions. In fact, one of the distinct four-bar linkages was missed by the homotopy continuation solver in [40] but was reconstructed using the cognate formula. Since a sharp upper bound has not yet been established, we aim to derive such an upper bound by considering a generalization of Alt’s problem which arises by counting the number of solutions outside of the base locus to a system of polynomials arising from a general linear combination of given polynomials.
Although formulating an upper bound on Alt’s problem in this fashion is new, the idea of constructing polynomial systems from randomly generated polynomials which have finitely-many solutions is not new. For example, the volume of Newton-Okounkov bodies [26, 27, 32] (which generalize the volume of Newton polytopes [9] in the monomial case) correspond with the number of solutions to such systems. Homotopies utilizing this structure have also been proposed [28] and comparisons between homotopy continuation and modular Gröbner basis methods have been performed [6]. Moreover, aspects of this problem have also been considered in the context of computing Euler characteristics [25, 29] and Segre classes [19, 24]. In addition to formulating an upper bound on Alt’s problem, the other novelty of our approach is to provide various probabilistic features of an effective algorithmic solution via modular Gröbner basis computations. We incorporate ideas used in the Gröbner trace algorithm [39] and other modular algorithms for Gröbner basis computation such as [4].
The rest of the paper is structured as follows. Section 2 formulates the problem and provides necessary background. Section 3 compares Hilbert functions computed using numerical methods and symbolic methods. Section 4 analyses the probability that randomly selected constants are general. Section 5 describes the results of some computational experiments and the paper ends with a summary of our results in the context of Alt’s problem in Section 6.
2. Formulation
For a system of polynomials in , say
[TABLE]
define
[TABLE]
For and an integer , we seek to compute
[TABLE]
where and are general, and is the length vector where each entry is . That is, we aim to count the number of solutions to a system of general affine linear polynomials and general linear combinations of outside of the base locus .
One approach for enforcing the solutions be outside of the base locus is via the Rabinowitz trick. That is, one introduces a new variable and, for general, we have
[TABLE]
Example 2.1**.**
To illustrate, consider consisting of polynomials in variables. Clearly, . Letting represent a general number,
[TABLE]
since there are points of intersection, all of which are away from the origin, between a general affine line and an irreducible quintic curve. Additionally,
[TABLE]
since the two irreducible quintic curves intersect in points, one of which is the origin. Since consists of monomials, these values can be computed via mixed volume computations, e.g., [9].
2.1. Relation to Alt’s problem
A four-bar linkage is the simplest moveable planar closed-chain linkage, one of which is shown in Figure 1 together with part of the coupler curve that it traces out. Without loss of generality, we can assign the origin [math] to be a point on the coupler curve and describe a four-bar linkage using points in the plane: , , , and . The points and correspond with the two fixed pivots while the segment is the so-called floating link. Since the planar points constitute design parameters, Alt realized that one can additionally specify general points resulting in finitely many four-bar linkages whose coupler curve passes through all nine specified points, i.e., the origin and others.
We proceed with a formulation inspired by [36, 40]. Utilizing isotropic coordinates, we write , , , and so that the eight design parameters are . A point on the coupler curve will be denoted . Isotropic coordinates yield a natural action applied to polynomials, denoted , that simply swaps and . In particular, given design parameters , the coupler curve traced out by the corresponding four-bar linkage is where
[TABLE]
with
[TABLE]
and
[TABLE]
[TABLE]
In fact, the map from design parameters to coupler curves is generically a -to- map. First, there is a trivial order action of relabeling, namely the design parameters
[TABLE]
yield the same coupler curve and actually describe the same mechanism. Additionally, there is an order action of Roberts cognates, e.g., see [40, Eq. 17], which yield distinct mechanisms that correspond to the same coupler curve.
Returning to Alt’s problem, suppose that for are general. Thus, one aims to find the design parameters such that, for , lies on its coupler curve. Hence, for and
[TABLE]
Alt’s problem of counting the number of distinct mechanisms is equal to one-half of the number of isolated points in while the number of distinct coupler curves is equal to one-sixth of the number of isolated points in . The homotopy continuation computation first described in [40] provides that the number of isolated points in is (at least) .
We obtain an upper bound on the number of isolated points in by replacing each coefficient with an independent parameter, say . The number of isolated solutions to the resulting system is exactly from (2) where . In fact, it is easy to verify that is the union of the following linear spaces:
[TABLE]
Note that each of these correspond to degenerate linkages that are not of interest. Therefore, is an upper bound on the number of distinct four-bar linkages whose coupler curve interpolates general points. Similarly, is an upper bound on the number of distinct coupler curves which interpolate general points. In addition, one naturally has a coefficient-parameter homotopy [30] for computing a superset of the isolated points of by simply deforming the coefficients to .
For , the number is an upper bound on the degree of the set of four-bar linkages whose coupler curve interpolates general points, a variety that has dimension . The degrees of these problems were first reported in [12, Table 1] and are displayed in Table 2. Similar to the results of [40] the computations in [12] do not prove that the degrees can be no larger than the values in Table 2 and we will use modular methods to provide a (probabilistic) confirmation of these values.
2.2. Relation to projective degrees and characteristic classes
For , we may also view each as the degree of the pullbacks of general linear spaces under a certain rational map. Consider the rational map
[TABLE]
and let be a general linear space passing through the origin of dimension in , i.e., having codimension in . Let be a matrix such that . Then,
[TABLE]
The number is also the volume of the Newton-Okounkov body [26, 27, 32] corresponding to .
More generally, if are subschemes inside of , which is a product of affine and projective spaces, and is a collection of polynomials in the coordinate ring of , we can consider the map
[TABLE]
In this case, we wish to compute the numbers
[TABLE]
If is a projective variety, then the polynomials making up are homogeneous, is replaced by , a set of polynomials defining can be taken to have the same degree (without changing ), and the numbers are the projective degrees of in (see [19] for the general case as well as [20, Example 19.4] and [24, 25] for the special case ). Moreover, it is shown in [24] that when is a subscheme of , the (pushforward of the) Chern-Schwartz-McPherson class in the Chow ring , namely , and hence the topological Euler characteristic, namely , are completely determined by appropriate projective degrees. Note that this also gives (the pushforward of) the Chern class of the tangent bundle, , since this agrees with whenever is smooth, i.e., whenever the Chern class is defined. For more details, see [2, §2.2]. Similarly, if are subschemes of a smooth projective toric variety , it is shown in [19] that the projective degrees completely determine the (pushforward of the) Segre class of in , namely , in the Chow ring . This Segre class in turn gives rise to many other invariants of potential interest such as Samuel’s algebraic multiplicity of along , namely , e.g., see [15, §4.3] and [19, §5], and polar classes, e.g., see [15, Ex. 4.4.5]. These Segre classes have also been used to give a Gröbner free test of pairwise containment of projective varieties, e.g., see [19, §6].
2.3. Plane conics
To illustrate several possible situations, we consider the enumerative geometry problem of counting the number of plane conics in passing through the origin and meeting general lines. First, we associate a plane conic passing through the origin with where
[TABLE]
We parameterize the lines by where are general for . Then, the enumerative geometry problem is to count the number of isolated points in , which is , where with
[TABLE]
and
[TABLE]
For and , we have
[TABLE]
which corresponds with planes that are not of interest. In fact, providing a sharp upper bound on the number of isolated points in .
Alternatively, we could work on the affine patch defined by , i.e., plane cubics defined by
[TABLE]
Hence,
[TABLE]
with and , which we will verify in Section 3.3.
Finally, one could work on , i.e., plane cubics defined by
[TABLE]
with
[TABLE]
so that
[TABLE]
which all correspond to degenerate cases that are not of interest, and .
3. Verification using Hilbert functions
For homogeneous ideals, modular computations can be utilized to provide upper bounds on the Hilbert function [4, Thm. 5.3]. However, since the ideals of particular interest are parameterized ideals which are generically radical and zero-dimensional but need not be homogeneous, this section compares affine Hilbert functions computed using numerical methods (yielding lower bounds) and symbolic methods (yielding upper bounds). In particular, when the upper and lower bounds agree, one has computed the generic value of the Hilbert function.
Consider the field consisting of rational functions in the parameters with rational coefficients. Let be the ring of polynomials in the variables whose coefficients are in . Fix . For a polynomial , let be the polynomial obtained by specializing the parameters to whenever all coefficients of are defined at . Suppose that generate an ideal that is generically radical and zero-dimensional. That is, for general values , with .
For , let and denote the vector space over of polynomials in and , respectively, of degree at most . In particular, . The affine Hilbert function of is the function defined by
[TABLE]
Moreover, for general ,
[TABLE]
3.1. Numerical Hilbert functions
With the setup described above, we develop a certified approach for computing lower bounds on the Hilbert function of that combines [17] with a method that guarantees the existence of a solution to a polynomial system, one such approach being -theory [10, Ch. 8]. We first introduce the exact computation followed by the certified numerical approach for computing lower bounds using a well-constrained system.
Suppose that and consists of nonsingular solutions. Let be the ideal of polynomials vanishing on . By the implicit function theorem, each point lifts locally to a solution in , say with .
Let be the degree Veronese embedding, i.e.,
[TABLE]
Moreover, let be
[TABLE]
With this setup, . The following theorem adds certification to produce guaranteed lower bounds on .
Theorem 3.1**.**
With the setup above, let , , and be a submatrix of . Then, provided that there exists a nonsingular solution to the well-constrained system
[TABLE]
where and is the identity matrix.
Proof.
Since is a submatrix of and both have rows,
[TABLE]
Suppose that is a nonsingular solution of . Note that is well-constrained consisting of polynomials in variables. Thus, the structure of yields that each is a nonsingular solution of and . Therefore,
[TABLE]
∎
In practice, one uses numerical approximations of known solutions to select and the corresponding submatrix which one expects to be invertible. Therefore, starting from numerical approximations, one can use a local certification routine, e.g., -theory [10, Ch. 8], to prove the existence of a nonsingular solution of thereby certifying .
For each , let be the largest value for which one can certify based on known numerical data. Hence, .
An upper bound on is developed in Section 3.2 with an example in Section 3.3.
3.2. Symbolic reduction
One option for computing the Hilbert function and thus is to compute a Gröbner basis for . However, when this is not practical, partial information provides upper bounds on the Hilbert function. For example, for each and , consider the linear space
[TABLE]
Clearly, showing that
[TABLE]
Moreover, and there exists such that for all showing that this upper bound eventually becomes sharp. In fact, one such stopping criterion is if such that , then for .
Let . Rather than taking an exhaustive approach that considers all possible polynomials, one could instead repeat similar computations on to the modular computations performed when computing a Gröbner basis for over a finite field. Thus, one uses these computations to guide the computation of the polynomials in via the Gröbner trace algorithm [39]. Since one is performing exact computations on , one maintains the upper bounds on .
3.3. Example
We demonstrate the lower and upper bounds via the parameterized system
[TABLE]
where is a matrix of parameters and is as in (7). Let and consider
[TABLE]
with . Using Bertini [7, 8], we computed numerical approximations of 18 points in . From this data, we applied Theorem 3.1 for to yield
[TABLE]
as follows. For , we selected points and utilized alphaCertified [23] with Theorem 3.1 to certify
[TABLE]
Therefore, we know .
For , we utilized all known points with the subset of columns corresponding to the following monomials:
[TABLE]
This certification via alphaCertified and Theorem 3.1 shows that . Since we utilized all points in this computation, we have for .
Next, we turn to upper bounds for for . The following tables summarize the exhaustive approach for symbolic reduction described in Section 3.2 for and various values of .
[TABLE]
Hence, . Combining with (9) shows that and thus for , i.e., . In particular, this shows that there are generically solutions to , i.e., where with as in (7).
4. Probabilistic analysis
The theory in the previous sections relied upon general choices of parameters. In this section, we give explicit characterization of when constants are sufficiently general for our computations. Since Alt’s problem is formulated affinely in Section 2.1, we focus on the affine case but note that the results proved here have analogs in the other settings discussed in Section 2.
Let , , and . Consider the ideal
[TABLE]
where , , and . The following is an affine version of [24, Thm. 4.1].
Theorem 4.1**.**
Let , , and . For general choices of , , and , as in (10) is either the unit ideal or has dimension [math]. In either case,
[TABLE]
is independent of , and (provided they are general).
Proof.
This immediately follows from Bertini’s Theorem and the Rabinowitz trick. The proof is identical to that of [24, Thm. 4.1] with replaced by . ∎
In practice, when computing the numbers , the constants , , and are chosen randomly from a finite set of rational numbers, e.g., all rational numbers of bounded height or a subset of floating-point numbers of a given bit size. Therefore, even working over , the computation of the integers in Theorem 4.1 is probabilistic. For our random choices in the following, we will utilize a uniform distribution on the chosen finite set.
Since the results below depend on generators with integer coefficients, we will assume that, without loss of generality, and that the coefficients in each are relatively prime.
4.1. Lucky primes for randomly constructed ideals
To simplify computations, we can employ modular methods which require the notion of lucky primes as in [4, §5.1] and [33].
The following is from [33, Defn. 4.1].
Definition 4.2**.**
For , let and be ideals with Gröbner bases and , respectively. A prime is Pauer lucky with respect to if and only if does not divide any of the leading coefficients of the polynomials in . Otherwise, is Pauer unlucky with respect to .
For and , let denote the set of leading monomials of . For a prime , let denote the leading monomials of . The following is [33, Prop. 4.1] which states that if a prime is Pauer lucky, then the ideal of leading terms computed over and agree.
Proposition 4.3** (Prop. 4.1 of [33]).**
With the setup as above, if a prime is Pauer lucky with respect to , then .
The following is used to avoid dividing any coefficients.
Definition 4.4**.**
For a finite collection of polynomials , we say that a prime is large relative to the coefficients of if is larger than any coefficient appearing in a polynomial contained in .
The following shows that every prime that is large relative to the coefficients of is Pauer lucky for defined in (10) on a Zariski dense set.
Theorem 4.5**.**
Following the notation from Theorem 4.1 such that , suppose that is a Zariski dense subset of where the statements of Theorem 4.1 hold. Also, suppose that for , we have that . Then, every prime which is large relative to the coefficients of is a Pauer lucky prime for the ideal on a Zariski dense set of parameters .
Proof.
Since each entry of , and is rational, we symbolically treat the numerator and denominator separately. To that end, let and denote the corresponding numerators and denominators of and , respectively. Let denote the collection of integer parameters.
For as in (10), let be the ideal obtained by clearing denominators in each of the generators of . Note that for any , the corresponding and define the same ideal in . Also note that since is large relative to the coefficients of , each coefficient in is not divisible by .
Given an integer vector where each entry of , , is nonzero, we let denote the corresponding rational values .
With this setup, we will consider as an ideal in the ring . Let be a Gröbner basis of consisting of, say, polynomials. Let be the leading coefficients of where each . Given and integer vector such that , the leading monomials of evaluated at are a superset of the leading monomials obtained by computing a Gröbner basis of in the ring . In particular, the evaluated values of the leading coefficients are multiples of the leading coefficients of a Gröbner basis of in the ring . Therefore, for every giving rise to a corresponding , either is Pauer lucky or Pauer unlucky. Hence, there must be a Zariski dense subset of such that at least one of these properties is satisfied. It is enough to show that there is a Zariski dense subset of such that for each in this subset with corresponding , we have that does not divide since this shows that does not divide the leading coefficients of a Gröbner basis of in the ring .
We note that if with corresponding , . In particular, each leading monomial of is not constant with respect to .
For the sake of reaching a contradiction, suppose that there exists a Zariski dense subset such that for all with corresponding integer vector , we have that is a Pauer unlucky prime for the ideal . If is Pauer unlucky for for all , then there must exist a fixed and a (possibly smaller) Zariski dense subset such that divides for all corresponding to .
Since is Zariski dense, it is also dense in the classical topology. This means that must be constant since a nonconstant polynomial must map dense sets to dense sets and the set of multiples of a prime is not dense in (in the classical topology). It follows that for all . This is not possible due to the construction of since every non-constant term of every polynomial in is multiplied by some parameter value, i.e., at least one parameter must appear in each leading term since the ideal is not generated by a constant. Hence, there is no dense subset of such that is Pauer unlucky. It follows that there must exist some dense subset of for which is a Pauer lucky prime for the ideal for all . ∎
We now illustrate the construction in the proof of Theorem 4.5 with an example.
Example 4.6**.**
Reconsider from Example 2.1 where with . Note that every prime is large relative to the coefficients of as in Definition 4.4. Since the ideal
[TABLE]
generically already has degree , we can simplify the presentation by just considering equal to the ideal in (12). Define
[TABLE]
so that is equivalent to
[TABLE]
Symoblically clearing denominators, we obtain the ideal , namely
[TABLE]
Using lexicographic order, the leading terms of a Gröbner basis for are
[TABLE]
where
[TABLE]
Generically, (13) yields that the standard monomials are . As expected, each leading coefficient is nonconstant.
For illustration, consider and corresponding with . Evaluating (13) yields the following leading terms:
[TABLE]
On the other hand, if we substitute and into (12), clear denominators, and compute a Gröbner basis for the resulting ideal in , the leading terms are
[TABLE]
As expected, each leading coefficient in (15) divides the corresponding coefficient in (14), e.g., and . For this problem, both (14) and (15) yield that the same set of Pauer unlucky primes via Definition 4.2, namely .
4.2. Probability of a prime being unlucky for randomly constructed ideals
Instead of considering a Zariski dense subset of as in Theorem 4.5, we now consider random choices from a finite subset.
Theorem 4.7**.**
Let , , and be a prime that is large relative to the coefficients of . Suppose that and each entry of is randomly selected from . Then, a lower bound on the probability that is a Pauer lucky prime for as in (10) is
[TABLE]
where is the number of polynomials in a Gröbner basis of in and
[TABLE]
Proof.
Consider as an ideal in the ring and let be a Gröbner basis of in this ring which consists of polynomials.
First, suppose the ideal in is not generated by a constant. From the proof of Theorem 4.5, we have that the leading coefficients of , say , are nonconstant polynomials in . Since a nonconstant polynomial must map to itself injectively, the probability that a given coefficient will be nonzero in where each entry of is chosen from is at least .
Since no leading monomial can have degree greater than and there are
[TABLE]
monomials having degree at most in variables, it follows that .
Finally, if is generated by a constant, i.e., an element of , the same lower bound on the probability of success holds. ∎
Remark 4.8**.**
Since the degree of is the quantity of interest, one could produce lower bounds on the probability in (16) by using any upper bound on the degree of , e.g., the multihomogeneous Bézout bound or the mixed volume. Further, note that [14, Cor. 2.1] yields the number of polynomials in a reduced Gröbner basis of a zero dimensional ideal over a field is at most the product of the number of variables and the degree of the ideal. While the ideal in Theorem 4.7 is zero dimensional, it is not computed over a field. However, it seems likely that there is a much lower upper bound on than the bound in Theorem 4.7.
In Theorem 4.7, each entry of is taken from a finite set while the ideal as in (10) is defined over . We now turn to treating as an ideal defined over a finite field. To that end, let denote the (correct) integer value as in (11) computed using general rational values for , , and as in Theorem 4.1. Let denote computed where the entries of , , and are randomly selected from a finite set . Finally, let denote where each entry of , , and is selected randomly among the elements of . The following provides a comparison of probabilities for computing the same number. In this statement, denotes the probability that random variables and take on the same value.
Theorem 4.9**.**
Following the setup of Theorem 4.7,
[TABLE]
where is as in Theorem 4.7.
Proof.
This follows immediately from Theorem 4.7. ∎
4.3. Probability of choosing general coefficients
Although Theorem 4.1 states that general values of the parameters yield the correct result, it does not provide insight into the set of parameters where (11) fails. The following provides degree bounds which are used to develop lower bounds on the probability of choosing general coefficients. We note that the results in this section are essentially an affine version of those in [25, §3.3, A.2].
As above, suppose that , , and . Define the projection of along as the rational map given by . Let be the value in (11) for general and let be the value obtained for a particular choice of .
Consider the algebraic sets
[TABLE]
We call the graph of . Define . In this notation, is simply the number of solutions (counted with multiplicity) to the system
[TABLE]
where and are general matrices.
We now treat and as variables and work in . We construct the corresponding discriminant variety as follows. Let
[TABLE]
and be the projection onto the and coordinates. Also, define
[TABLE]
and set
[TABLE]
The discriminant (as shown below in Proposition 4.11) is given by
[TABLE]
that is, we show that for , we have that . The following helps to show this.
Lemma 4.10**.**
Let . Suppose that
[TABLE]
and let . Then, all points in are such that and the map gives a bijection with inverse map .
Proof.
Note that may be defined as the Zariski closure of the image of the restriction of to either or since these are the same. Let . Since , we can set and hence . It follows that and is thus contained in .
By assumption, so take to be a point lying over . Since , this implies that (since and must be disjoint and ). Hence, which implies that . ∎
Proposition 4.11**.**
Using the notation above, for .
Proof.
The proof is split into two cases: either the restriction of to is surjective or not.
Suppose that restricted to is not surjective. Note that since the map is closed, then we must have that has codimension at least one. If , then we have that there is no such that . Note that by definition. By Lemma 4.10, it follows that is empty. Hence, .
Suppose that restricted to is surjective. Let . By [25, Lemma A.11], we have that is a regular value of the restriction of to if and only if for any in the fiber , we have that . Hence, for we have that is a regular value of the restriction of to . Then, since restricted to is surjective, it follows that is a dominant map, and hence the fibers of its regular values have the same cardinality (e.g., see [18, Prop. 12.19, Cor. 12.20, (12.6.2)]). The conclusion follows by Lemma 4.10. ∎
The following provides a degree bound.
Proposition 4.12**.**
Let , , and . Then, there exists a polynomial of degree at most
[TABLE]
such that if , then .
Proof.
We first consider the case where restricted to is surjective. Note that we may choose the polynomial of minimal degree to be when defining , i.e., assume . We must bound the degree of the discriminant . By Bézout’s Theorem,
[TABLE]
We next bound . Let . For , we have . It follows that is one of the irreducible components of . Hence, . Moreover,
[TABLE]
showing that . Therefore, . Since degrees cannot increase under projection, it follows that .
Next, we need to bound . The polynomial has degree at most and . The result now follows from (20).
The nonsurjective case follows similarly with . ∎
With this upper bound on degree, we can derive probabilistic bounds.
Proposition 4.13**.**
With the setup from Proposition 4.12, if the values for , , and are chosen randomly from a finite set of , then
[TABLE]
Proof.
Consider the ideal
[TABLE]
Let be as in Proposition 4.12 and suppose and are selected such that . Then, we know and . Suppose that and consider the following polynomial of degree :
[TABLE]
Note that if , then . The probability bound follows from applying the Schwartz-Zippel Lemma [37, 41] using the polynomial . ∎
The following considers the computation over finite fields.
Corollary 4.14**.**
Let , , and be a prime that is large relative to the coefficients of . Set , , and
[TABLE]
Suppose that
[TABLE]
is computed over where each entry of , , and are selected randomly from . Then,
[TABLE]
where is as in Theorem 4.7 with the upper bound
[TABLE]
Proof.
This follows immediately from Theorem 4.9 and Proposition 4.13. ∎
5. Computations
The following considers problems associated with Alt’s problem introduced in Section 2.1. We let as in Section 2.1 which depend upon variables with as in (3). Throughout this section, we will suppose that all primes being considered are such that so that is large relative to the coefficients of as in Definition 4.4. The following finds primes that are large enough to provide a high probability that . The modular computations described below utilized Magma [11].
5.1. Probabilistic considerations
The following specialize results from Section 4 to the setup related to Alt’s problem for computing the numbers . In particular, we provide theoretical bounds on the size of the prime required to obtain a desired level of certainty in computing using random choices with computations over .
Corollary 5.1**.**
With the setup described above,
[TABLE]
where as in Theorem 4.7 which is bounded above by .
Proof.
Using the notation of Corollary 4.14, we have , , , and . Using Bezóut’s theorem, we obtain the bound . ∎
For example, when , is generated by eight general linear combinations of along with the polynomial . The results of [31] can be extended to this situation to yield . For , this gives
[TABLE]
For example, if we take , we have Although this bound is pessimistic, it provides an a priori lower bound on the probability. The following compares the probabilistic bound for with what is actually achieved in experiments.
5.2. Practical success rate for
To compare the theoretical lower bounds with the practical success rate, we consider the simplest nontrivial case, namely computing . Bézout’s Theorem together with described in (3) provides while the actual value of can be verified using methods described in Section 3. Using with Corollary 5.1, we have
[TABLE]
Therefore, the theoretical results show that will yield a probability of success over . For this problem, note that the size of is dominated by the second term of the product since
[TABLE]
To check the practical success rate for various smaller primes , we performed 10,000 random trials for selected primes . Table 1 lists the number of successes, i.e., when . In particular, this experiment, as expected, shows that the a priori theoretical bounds are quite pessimistic. For example, for all the selected primes , we already obtained a success rate over in 10,000 random trials.
5.3. Larger computations
As mentioned in Section 2.1, is an upper bound on the degree of the set of four-bar linkages whose coupler curve interpolates general points. The actual degree of these sets were first reported in [12, Table 1] which were computed using homotopy continuation. For convenience, we lists these degrees in Table 2.
Trivially, . For , the results of Section 5.2 show that is also a sharp upper bound. Hence, we consider the sharpness for by computing for various primes where the parameters are selected uniformly at random in . The results of these computations, which were performed using a 2.5 GHz Intel Xeon E5-2680 v3 processor with 256 GB of memory, are summarized in Table 3. The ones marked with “” failed to finish in 10 days, while the ones in bold agree with the degrees reported in Table 2. In particular, these results suggest that is indeed a sharp upper bound for all . We note that Table 3 shows a considerable increase in the computational time needed for the two largest primes in our experiment compared with the other selected primes.
To provide additional validation of , we performed 10 additional trails of for as well as using randomly selected floating-point parameters via homotopy continuation in Bertini. All of these experiments also yielded .
Table 4 lists the purported Hilbert functions in . These are in agreement with numerical computations based on using solutions computed by homotopy continuation in Bertini.
6. Conclusion
For a polynomial system with and a variety containing , we considered the computational problem of computing for as defined in (6). This question was primarily motivated by its application to providing a sharp upper bound to Alt’s problem, namely where as formulated in Section 2.1. More generally, the values also arise in the computation of the volumes of Newton-Okounkov bodies and for the computation of characteristic classes such as Chern and Segre classes as described in Section 2.2. In particular, Section 4 provides an analysis of a probabilistic saturation technique for computing using Gröbner basis computations over finite fields.
In the context of Alt’s problem, the numbers for provide an upper bound on the degree of the set of four-bar linkages whose coupler curve interpolates general points, which is a variety of dimension . In fact, the solution to Alt’s original problem from 1923 on the number of distinct four-bar linkages that interpolate general points is bounded above by and the number of distinct coupler curves that interpolate general points is bounded above by . In 1992, homotopy continuation was used in [40] to show the actual numbers were 4326 and 1442, respectively. However, due to the nature of the computation, they could not be sure that no additional solutions could exist. In fact, it is reported in [40] that one of the solutions was missed by their homotopy continuation solver, but was reconstructed using Roberts’ cognate formula. Therefore, we posit that verifying theoretically completes Alt’s problem by showing that there could be no additional solutions. To that end, we analyzed a probabilistic saturation technique (see Theorem 4.1 and Corollary 4.14) using Gröbner basis computations over various finite fields to show for various primes , a result that is also confirmed numerically using homotopy continuation.
As shown in Section 5.1, we were able to derive a priori theoretical bounds from Corollary 4.14 for the probabilistic computation of . This bound requires the prime characteristic of our finite field to be in order to guarantee a probability of success greater than . Unfortunately, our computational resources together with Magma do not currently permit computing for . In fact, Table 3 shows a drastic increase in computational time between and .
On the other hand, Section 5.2 shows the pessimism of the a priori theoretical bounds in that much higher success rates can be obtained in practice when computing over finite fields with much smaller characteristic. In particular, for the probabilistic computation of , the bound of Corollary 4.14 suggests one would need to take a finite field with prime characteristic larger than to obtain a success rate greater than . Experimentally, we found that we achieved this success rate on 10,000 random trials with all selected primes . Further, even with the prime , we achieved a success rate of over in our experiment. Naturally, this leads us to conclude that the bound of Corollary 4.14 of for computing with a success rate greater than is also overly pessimistic. Coupled with numerical homotopy continuation computations, we have confidence that the probabilistic computational results indeed correctly yield .
Finally, we remark that the probabilistic saturation technique has permitted computations related to Alt’s problem to terminate using symbolic methods. For example, our probabilistic saturation computations related to completed on average in under seconds for our experiments using both Magma and Macaulay2 [16] for the primes listed in Table 1. A similar problem was attempted in [6, §5.4] using primary decomposition over finite fields via Singular [13], a computation which failed to terminate in 24 hours. In fact, to generate the data for Table 1, the 10,000 random trials for each prime were completed in under 10 minutes. This drastic increase in speed allowed us to successful complete random trials for computing associated with Alt’s problem for for various primes as reported in Section 5. In particular, since the values of match the results in Table 2, we have utilized symbolic methods to confirm numerical homotopy continuation computations in [12, 40].
Acknowledgments
The authors would like to thank the Fall 2018 semester program on Nonlinear Algebra at the Institute for Computational and Experimental Research in Mathematics (ICERM) where this project was started. JDH was supported in part by NSF CCF 1812746.
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