Computing Circuit Polynomials in the Algebraic Rigidity Matroid

The focus of this paper is the following problem straddling combinatorial rigidity and algebraic matroids.

Main Problem. Given a rigidity circuit, compute its corresponding circuit polynomial.

Its motivation comes from the following ubiquitous problem in distance geometry.

Localization. A graph together with weights associated to its edges is given. The goal is to find placements for its vertices in some Euclidean space (2D, in our case), so that the resulting edge lengths match the given weights. To this purpose we set up a system of quadratic equations with unknowns corresponding to the Cartesian coordinates of the vertices. The possible placements (or realizations) are among its (real) solutions and can be found with numerical methods (see, e.g., [36, 49, 3]). A related problem is to look for the possible values of a single unknown distance corresponding to a nonedge (a pair of vertices that are not connected by an edge). If we could solve this second problem for a collection of nonedge pairs that, together with the original edges, contain a trilateration, then one placement for the graph could be obtained afterwards in linearly many steps of quadratic equation solving.

Rigidity circuits. The generic version of the single unknown distance problem, where the weights are symbols rather than concrete numbers, is amenable to techniques from rigidity theory. In 2D, one can predict whether, generically, the set of solutions for the unique unknown distance will be discrete (if the given graph is rigid) or continuous (if the graph is flexible). We formulate the problem algebraically by using Cayley coordinates \(X_n = \{x_{ij}: 1\leq i \lt j\leq n\}\) , with \(x_{ij}\) denoting the squared distance between vertices \(i\) and \(j\) , and \(n\) being the number of vertices. There are certain dependencies between these variables, captured by the polynomials \(f\in \mathbb{Q}[X_n]\) generating the Cayley–Menger ideal. When \(G\) is a minimally rigid graph, the addition of a new edge \(e\) induces a unique subgraph \(C\subseteq G\cup \{ e\}\) , which is a circuit in the 2D rigidity matroid whose bases are the minimally rigid graphs. There also exists a unique (up to multiplication by a scalar) polynomial dependency \(p_C\) between the distances corresponding to the edges of \(C\) . This is a circuit polynomial in the Cayley–Menger ideal and is the main object of study in this paper. The unique unknown distance problem is solved by substituting in this circuit polynomial concrete values for the edge weights of \(G\) and then computing the roots of the resulting univariate polynomial.

How tractable is the problem? Circuit polynomial computations can be done, in principle, by using the Gröbner basis algorithm with an elimination order.¹ In the worst case, this is a doubly exponential method, but in practice the complexity and performance of Gröbner basis algorithms depends heavily on the choice of a monomial order. There exist known cases, e.g., zero-dimensional polynomial ideals [15, 32], which have single-exponential complexity with respect to any monomial order. However, elimination orders have been reported to behave badly. In general, the main problems of elimination theory, such as the Ideal Triviality Problem, the Ideal Membership Problem for Complete Intersections, the Radical Membership Problem, the General Elimination Problem, and the Noether Normalization, are in the PSPACE complexity class [40].

In our experimentation, the GroebnerBasis function of Mathematica 12 (running on a 2019 iMac computer with 6 cores at 3.6 Ghz) took 5 days and 6 hours to compute the Desargues-plus-one circuit (a graph on 6 vertices) reported in Table 1 of section 13, but in most cases it timed out or crashed.

Overview of results. Our goal is to make such calculations more tractable by taking advantage of structural information inherent in the problem. We describe a new algorithm to compute a circuit polynomial with known support. It relies on resultant-based elimination steps guided by a novel inductive construction for rigidity circuits. Inductive constructions have been often used in rigidity theory, most notably the Henneberg sequences for Laman graphs [27] and Henneberg II sequences for 3-connected rigidity circuits [5]. We argue that our combinatorial construction is more natural due to its direct algebraic interpretation, a property not shared with any of the other previously known constructions. We have implemented our method in Mathematica and applied it successfully to compute all but one of the circuit polynomials on up to 6 vertices, as well as a few on 7 and 8 vertices, the largest of which having over nine million terms. The previously mentioned example of the Desargues-plus-one circuit that took over 5 days to complete with GroebnerBasis, was solved by our algorithm in less than 15 seconds.

The only example on 6 vertices that remained elusive was the circuit polynomial for the \(K_{3,3}\) -plus-one circuit (see Table 1 in section 13): the computational resources for its computation far exceeded the capabilities of both our machines and of an HPC system we experimented with. We succeeded by extending the basic algorithm to work with additional generators of the Cayley–Menger ideal, besides those corresponding to \(K_4\) ’s. These are irreducible polynomials supported on dependent rigid graphs that are not necessarily circuits.

Related work. Our approach builds upon ideas from distance geometry and rigidity theory and combines them with the theory of algebraic matroids. The former enjoy a long and distinguished history—too long to survey here, but see [6, 14]. Combinatorial and linear (but not algebraic) matroids occupy a central place in rigidity theory [24, 55]. To the best of our knowledge, the study of circuit polynomials in arbitrary polynomial ideals was initiated in the Ph.D. thesis of Rosen [45]. His Macaulay2 code [46] is useful for exploring small cases, but the Cayley–Menger ideal is beyond its reach. A recent article [47] popularizes algebraic matroids and uses for illustration the smallest circuit polynomial \(K_4\) in the Cayley–Menger ideal. We could not find nontrivial examples anywhere. Indirectly related to our problem are results such as [54], where an explicit univariate polynomial of degree 8 is computed (for an unknown angle in a \(K_{3,3}\) configuration given by edge lengths, from which the placement of the vertices is determined), and [48], for its usage of Cayley coordinates in the study of configuration spaces of some families of distance graphs. A closely related problem is that of computing the number of embeddings of a minimally rigid graph [9], which has received a lot of attention in recent years (see, e.g., [11, 1, 19, 18], to name a few). References to specific results in the literature that are relevant to the theory developed here and to our proofs are given throughout the paper.

Overview of the paper. Our main theoretical result is split into a combinatorial theorem, Theorem 1, and an algebraic theorem, Theorem 2, each with an algorithmic counterpart and each preceded by a section introducing the concepts necessary for a self-contained presentation. Section 2 reviews 2D combinatorial rigidity matroids. Then in section 3 we define the combinatorial resultant of two graphs as an abstraction of the classical resultant, prove Theorem 1, and describe the algorithm for computing a combinatorial circuit-resultant (CCR) tree.

This leads to a graph algorithm for finding a CCR tree of a circuit. Each step of the construction can be carried out in polynomial time using variations on the pebble game matroidal sparsity algorithms [35] combined with Hopcroft and Tarjan’s linear time 3-connectivity algorithm [28]. However, it is conceivable that the tree could be exponentially large, and thus the entire construction could take an exponential number of steps: understanding in detail the algorithmic complexity of our method remains a problem for further investigation.

In sections 4, 5, 6, and 7 we include a brief, self-contained overview of the algebraic concepts relevant to this paper: ideals and their algebraic matroids, the Cayley–Menger ideal, resultants, and the circuit polynomials in the Cayley–Menger ideal. In section 8 we prove the following theorem.

The algorithmic counterpart of Theorem 2 appears in section 9. Overall, the resulting algebraic elimination algorithm runs in exponential time, in part because of the growth in size of the polynomials that are being produced. Several theoretical open questions remain, whose answers may affect the precise time complexity analysis.

In section 10 we define and characterize a more general combinatorial resultant tree which generalizes the CCR tree by allowing more freedom in the choice of graphs used at the leaves of the tree: besides \(K_4\) circuits, we now can use dependent rigid graphs. This extension allows the use of polynomials supported on dependent sets in the Cayley–Menger ideal that are not necessarily circuits. The dependent, noncircuit generators of the Cayley–Menger ideal are discussed in section 11, and the full generalization of our main algorithm is given in section 12.

The preliminary experimental results we carried out with the implementation of our method in Mathematica are discussed in section 13. We used Mathematica v13 on an 2019 iMac with the following specifications: Intel i5-9600K 3.7GHz, 16GB RAM, macOS Monterey 12.3.1. We also explored Macaulay2, but it was much slower than Mathematica (hours vs. seconds) in computing one of our examples. The resulting polynomials are made available on a GitHub repository [39].

Open questions are introduced throughout the paper and in the concluding remarks, section 14.

Further connections: Circuit polynomials in matroid theory. The matroid theory literature is rich in realizability questions of various sorts [43] and has seen in recent years a surge of interest in algebraic matroids. Ingleton [29] proved that algebraic matroids over fields of characteristic 0 are linearly realizable, but this is not the case in positive characteristic [43]. Recently, [7] identified an infinite class of algebraic matroids over fields of positive characteristic that have a linear representation in the same characteristic, namely, those for which the so-called Lindström valuation is trivial. The problem of computing the Lindström valuation was addressed in [12], where the fundamental step is to compute all circuit polynomials of a given algebraic matroid in positive characteristic. We remark that for the algebraic matroids whose combinatorial structure allows descriptions of their circuits in terms of an operation similar to our combinatorial resultants, the methods presented in this paper are applicable and likely to be more efficient than Gröbner basis methods.

We start with the combinatorial aspects of our problem and review the relevant notions and results from combinatorial rigidity theory of bar-and-joint frameworks in dimension 2.

Notation. We work with (sub)graphs given by subsets \(E\) of edges of the complete graph \(K_n\) on vertices \([n]:=\{1,\dots,n\}\) . If \(G\) is a (sub)graph, then \(V(G)\) , respectively, \(E(G)\) , denotes its vertex set, respectively, edge set. The support of \(G\) is \(E(G)\) . The vertex span \(V(E)\) of edges \(E\) is the set of all edge-endpoint vertices. A subgraph \(G\) is spanning if its edge set \(E(G)\) spans \([n]\) . The neighbors \(N(v)\) of vertex \(v\) are the vertices adjacent to \(v\) in \(G\) .

Frameworks. A 2D bar-and-joint framework is a pair \((G,p)\) of a graph \(G=(V,E)\) and a placement map \(p: V\to \mathbb R^2\) . We view the edges as rigid bars and the vertices as rotational joints which allow the framework to deform continuously as long as the bars retain their original lengths. The realization space of the framework is the set of all of its possible placements in the plane with the same bar lengths. Two realizations are congruent if they are related by a planar isometry. The configuration space of the framework is made of congruence classes of realizations. The deformation space of a given framework \((G,p)\) is the connected component of the configuration space that contains this particular placement (given by \(p\) ). A framework is rigid if its deformation space consists of exactly one configuration, and is flexible otherwise. We say that a framework is minimally rigid if it is rigid and, when any of its edges is removed, it becomes flexible.

Laman graphs. The concept of a generic framework is introduced rigorously in section 5. All but a measure-zero set of possible placements of a graph are generic. The following theorem allows us to refer to the rigidity and flexibility of a generic framework solely in terms of its underlying graph. The proof goes through the intermediate concept of infinitesimal rigidity, which implies rigidity; this is also introduced in section 5.

A graph satisfying the conditions of Theorem 3 is said to be a Laman graph, or just Laman. The hereditary property (b) is also referred to as the \((2,3)\) -sparsity condition. Together, properties (a) and (b) define a graph said to be \((2,3)\) -tight (in addition to being \((2,3)\) -sparse).

Theorem 3 allows us to talk now about (minimal) rigidity of graphs rather than frameworks. A Laman graph is minimally rigid and becomes flexible when any of its edges is removed. Adding extra edges to a Laman graph keeps it rigid, but the minimality is lost: these graphs are said to be rigid and overconstrained or dependent. In short, for a graph to be rigid, its vertex set must span a Laman graph; otherwise the graph is flexible. Other graphs may be simultaneously flexible and overconstrained. In this paper, we work primarily with graphs which are rigid and dependent. The minimally dependent ones, called rigidity circuits, are introduced next.

Matroids. A matroid is an abstraction capturing (in)dependence relations among collections of elements from a ground set and is inspired by both linear dependencies (among, say, rows of a matrix) and algebraic constraints imposed by algebraic equations on a collection of otherwise free variables. The standard way to specify a matroid is via its independent sets, which have to satisfy certain axioms (which we omit, and refer the interested reader to [43]). A base is a maximal independent set, and a set which is not independent is said to be dependent. A minimal dependent set is called a circuit. Relevant for our purposes are the following general aspects: (a) (hereditary property) a subset of an independent set is also independent; (b) all bases have the same cardinality, called the rank of the matroid. Further properties will be introduced in context, as needed.

In this paper we encounter three types of rigidity-related matroids: a graphic² matroid, defined on a ground set given by all the edges \(E_n:=\{ij: 1\leq i \lt j \leq n\}\) of the complete graph \(K_n\) ; this is the \((2,3)\) -sparsity matroid or the generic 2D rigidity matroid described below; a linear matroid, defined on an isomorphic set of row vectors of the rigidity matrix associated to a bar-and-joint framework; and an algebraic matroid, defined on an isomorphic ground set of variables \(X_n:=\{x_{ij}: 1\leq i \lt j \leq n\}\) ; this is the algebraic matroid associated to the Cayley–Menger ideal. The linear and algebraic matroids will be defined in section 5.

The \((2,3)\) -sparsity matroid: Independent sets, bases, circuits. The \((2,3)\) -sparse graphs on \(n\) vertices form the collection of independent sets for a matroid \(\operatorname{\mathbf{\mathcal S}}_{n}\) on the ground set \(E\) of edges of the complete graph \(K_n\) [55], called the (generic) 2D rigidity matroid, or the \((2,3)\) -sparsity matroid. The bases of the matroid \(\operatorname{\mathbf{\mathcal S}}_{n}\) are the maximal independent sets, and hence are Laman graphs. A set of edges which is not sparse is a dependent set. For instance, adding one edge to a Laman graph creates a dependent set of \(2n-2\) edges, called a Laman-plus-one graph; examples are given in Figure 1.

A minimal dependent set is a (sparsity) circuit. The edges of a circuit span a subset of the vertices of \(V\) . A circuit spanning \(V\) is said to be a spanning or maximal circuit in the sparsity matroid \(\operatorname{\mathbf{\mathcal S}}_{n}\) . See Figure 1 (right) and Figure 2 for examples.

A Laman-plus-one graph contains a unique subgraph which is minimally dependent, in other words, a unique circuit. A spanning rigidity circuit \(C=(V,E)\) is a special case of a Laman-plus-one graph: it has a total of \(2n-2\) edges but it satisfies the \((2,3)\) -sparsity condition on all proper subsets of at most \(n^{\prime}\leq n-1\) vertices. Simple sparsity considerations can be used to show that the removal of any edge from a spanning circuit results in a Laman graph.

Combining graphs and circuits. We define now operations that combine two graphs (with some common vertices and edges) into one.

If \(G_1\) and \(G_2\) are two graphs, we use a consistent notation for their number of vertices and edges \(n_i=|V(G_i)|\) , \(m_i=|E(G_i)|\) , \(i=1,2\) , and for their union and intersection of vertices and edges, as in \(V_{\cup }=V(G_1)\cup V(G_2)\) , \(V_{\cap }=V(G_1)\cap V(G_2)\) , \(n_{\cup }=|V_{\cup }|\) , \(n_{\cap }=|V_{\cap }|\) , and similarly for edges, with \(m_{\cup }=|E_{\cup }|\) and \(m_{\cap }=|E_{\cap }|\) . The common subgraph of two graphs \(G_1\) and \(G_2\) is \(G_{\cap } = (V_{\cap }, E_{\cap })\) .

Let \(G_1\) and \(G_2\) be two graphs with exactly two vertices \(u, v\in V_{\cap }\) and one edge \(uv \in E_{\cap }\) in common. Their 2-sum is the graph \(G=(V,E)\) with \(V=V_{\cup }\) and \(E=E_{\cup } \!\setminus\! \{uv\}\) . The inverse operation of splitting \(G\) into \(G_1\) and \(G_2\) is called a 2-split or 2-separation (Figure 3).

Connectivity. It is well known and easy to show that a circuit is always a 2-connected graph. If a circuit is not 3-connected, we refer to it simply as a 2-connected circuit. The Tutte decomposition [51] of a 2-connected graph into 3-connected components amounts to identifying separating pairs of vertices. For a circuit, the separating pairs induce 2-split (inverse of 2-sum) operations and produce smaller circuits (see also Lemma 2.4(c) in [5]). Thus a 2-connected circuit can be constructed from 3-connected circuits via 2-sums, as illustrated in the right-to-left sequence from Figure 3.

Inductive constructions for 3-connected circuits. A Henneberg II extension (also called an edge splitting operation) is defined for an edge \(uv\) and a nonincident vertex \(w\) , as follows: the edge \(uv\) is removed, and a new vertex \(a\) and three new edges \(au, av, aw\) are added. Berg and Jordán [5] have shown that if \(G\) is a 3-connected circuit, then a Henneberg II extension on \(G\) is also a 3-connected circuit. The inverse Henneberg II operation on a circuit removes one vertex of degree 3 and adds a new edge among its three neighbors in such a way that the result is also a circuit; see Figure 4. Berg and Jordan have shown that every 3-connected circuit admits an inverse Henneberg II operation which also maintains 3-connectivity. As a consequence, a 3-connected circuit has an inductive construction, i.e., it can be obtained from \(K_4\) by Henneberg II extensions that maintain 3-connectivity. Their proof is based on the existence of two nonadjacent vertices with 3-connected inverse Henneberg II circuits. We will make use in section 3 of the following weaker result, which does not require maintaining of 3-connectivity in the inverse Henneberg II operation.

We define now a new operation, the combinatorial resultant of two graphs, prove Theorem 1, and describe its algorithmic implications.

We turn now to the algebraic aspects of our problem in order to introduce algebraic matroids and circuit polynomials. We work over the field of rational numbers \(\mathbb{Q}\) . In this section, the set of variables \(X_n\) denotes \(X_n = \{x_i: 1\leq i\leq n\}\) ; when we turn to the Cayley–Menger ideal, it will be \(X_n = \{x_{ij}: 1\leq i \lt j \leq n\}\) . Polynomial rings \(R\) are always of the form \(R=\mathbb{Q}[X]\) , over sets of variables \(X\subset X_n\) . The support \(\operatorname{supp}{f}\) of a polynomial \(f\in \mathbb{Q}[X_n]\) is the set of indeterminates appearing in it. The degree of a variable \(x\) in a polynomial \(f\) is denoted by \(\deg_xf\) .

In this section we introduce the 2D Cayley–Menger ideal \(\operatorname{CM}_{n}\) . We will show³ that its algebraic matroid is isomorphic to the \((2,3)\) -sparsity matroid \(\operatorname{\mathbf{\mathcal S}}_{n}\) . As a consequence, we get a full combinatorial characterization of the supports of circuit polynomials in the Cayley–Menger ideal: they are in one-to-one correspondence with the rigidity circuits introduced in section 2.

Throughout this section and later, when working with the Cayley–Menger ideal, we use variables \(X_n = \{x_{i,j}: 1\leq i \lt j\leq n\}\) for unknown squared distances between pairs of points.

In this section we review known concepts and facts about resultants; in the next section we specialize this setup to the Cayley–Menger ideal. In section 8, in order to prove Theorem 2, we will use the resultant of two circuit polynomials in the Cayley–Menger ideal as the algebraic counterpart of the combinatorial resultant operation which deletes a common edge \(e\) of two circuits.

Resultants. The resultant can be introduced in several equivalent ways [22]. Here we use its definition as the determinant of the Sylvester matrix.

Let \(f,g\in R[x]\) be two polynomials in \(x\) with coefficients in some ring of polynomials \(R\) , with \(\deg_xf=r\) and \(\deg_xg=s\) , such that at least one of \(r\) or \(s\) is nonzero, and letThe resultant of \(f\) and \(g\) with respect to the indeterminate \(x\) , denoted \(\operatorname{Res}(f, g, x)\) , is the determinant of the \((r+s)\times (r+s)\) Sylvester matrix made from the coefficients of \(f\) and \(g\) arranged in staggered rows according to the following patternwhere the submatrix \(S_f\) containing only the coefficients of \(f\) is of dimension \(s\times (r+s)\) , and the submatrix \(S_g\) containing only the coefficients of \(g\) is of dimension \(r\times (r+s)\) . Unless \(r=s\) , the columns \((a_{0}\;a_{1}\;\cdots \;a_r)\) and \((b_{0}\;b_{1}\;\cdots \;b_s)\) of \(S_f\) and \(S_g\) , respectively, are not aligned in the same column of \(\operatorname{Syl}(f,g,x)\) , as displayed above, but rather the first is shifted to the left or right of the second, depending on the relationship between \(r\) and \(s\) . We will make implicit use of the following well-known symmetric and multiplicative properties of the resultant.

The first two properties can be found in [22, p. 398]. The third one is stated, without proof, in [25, p. 9] for unique factorization domains. When \(R\) is a field, a proof of this property can be found in [13, Chapter 3, Proposition 3 of section 6], and it directly generalizes to polynomial rings via Hilbert’s Nullstellensatz.

Resultants and elimination ideals. We will work with multivariate homogeneous polynomials \(f\) and \(g\) in \(\mathbb{Q}[X_n]\) , where a particular variable \(x\in X_n\) is singled out. Since the resultant is a polynomial in the coefficients of \(f\) and \(g\) , its net effect is that the specific variable \(x\) is being eliminated. Formally, let \(X^{\prime}\subset X\) be nonempty and let \(R=\mathbb Q[X^{\prime}]\) . Let \(f,g\in R[x]\) , where \(x\in X\!\setminus\! X^{\prime}\) . It is clear from the definition of the resultant that \(\operatorname{Res}(f, g, x)\in R\) . We will make frequent use of the following proposition, summarizing this observation; its proof can be found in [13, p. 167].

Homogeneous properties. From the next section on we will be working in the Cayley–Menger ideal, where the generators and the circuit polynomials are homogeneous. In sections 8 and 13 we will make use of the following proposition.

We were not able to find a reference for this proposition in the literature. In [13, p. 454] (Lemma 5 of section 7 of Chapter 8) we found the following special case: let \(f\) and \(g\) be homogeneous polynomials of degree \(r\) , respectively, \(s\) , with \(\deg_xf=r\) and \(\deg_xg=s\) , so that \(f=a_{0}x^{r}+\cdots +a_{1} x+a_r\) and \(g =b_{0}x^{s}+\cdots +b_{1} x+b_s\) . In this case \(\operatorname{Res}(f, g, x)\) is of homogeneous degree \(rs\) . The proof below is a direct adaptation of the proof of this special case, which itself follows directly from Proposition 23 by substituting \(m\to r\) and \(n\to s\) so to obtain \(rs+sr-rs=rs\) .

In this section we define circuit polynomials in the Cayley–Menger ideal and make the connection with combinatorial rigidity circuits via their supports.

Circuits of \(\rm {\operatorname{\mathbf{\mathcal A}}(\operatorname{CM}_{n})}\) and circuit polynomials in \(\operatorname{CM}_{n}\) . The isomorphism between the algebraic matroid \(\operatorname{\mathbf{\mathcal A}}(\operatorname{CM}_{n})\) and the sparsity matroid \(\operatorname{\mathbf{\mathcal S}}_{n}\) (Theorem 20) immediately implies that the sets of circuits of these two matroids are in a one-to-one correspondence. We will identify a sparsity circuit \(C=(V_C,E_C)\in \operatorname{\mathbf{\mathcal S}}_{n}\) , with the algebraic circuit \(\{x_{i,j}\mid ij\in E_C\} \in \operatorname{\mathbf{\mathcal A}}(\operatorname{CM}_{n})\) , and similarly for dependent sets. Conversely, we will identify the support of a polynomial \(f\in \mathbb{Q}[\{x_{i,j}\mid 1\leq i \lt j\leq n\}]\) with the graph \(G_f=(V_f,E_f)\) , where

Given a (rigidity) circuit \(C\) , we denote by \(p_C\) the corresponding circuit polynomial in the Cayley–Menger ideal \(\operatorname{CM}_{n}\) . Recall that by Theorem 18 the circuit polynomial of a circuit \(C\) in \(\operatorname{CM}_{n}\) is the unique (up to multiplication with a unit) polynomial \(p_C\) irreducible over \(\mathbb Q\) such that \(\operatorname{supp}{p_C}=C\) . Hence we will identify from now on a circuit \(C\) with the support \(\operatorname{supp}{p}_C\) of its circuit polynomial \(p_C\) . Furthermore, \(p_C\) generates the elimination ideal \(\operatorname{CM}_{n}\cap \ \mathbb{Q}[C]\) .

Example: The \(\rm K_4\) circuit. The smallest circuit polynomials are found among the generators of \(\operatorname{CM}_{n}\) . Their supports are in correspondence with the edges of complete graphs \(K_4\) on all subsets of 4 vertices in \([n]\) . The circuit polynomial \(p_{K_4^{1234}}\) given below corresponds to a \(K_4\) on vertices 1234. It is homogeneous of degree 3, has 22 terms, and has degree 2 in each of its variables:

Resultants of circuit polynomials. Let \(f,g\) be two polynomials in the Cayley–Menger ideal with \(x_{ij}\) one of their common variables. We treat them as polynomials in \(x_{ij}\) , and therefore the coefficients are themselves polynomials in the remaining variables. Our main observation, which motivated the definition of the combinatorial resultant, is that the entries in the Sylvester matrix are polynomials supported exactly on the variables corresponding to the combinatorial resultant of the supports of \(f\) and \(g\) on elimination variable (edge) \(ij\) .

The following lemma, whose proof follows immediately from Proposition 22, will be used frequently in the rest of the paper.

We are now ready to complete the proof of our second result, Theorem 2. We show that combinatorial resultants are the combinatorial analogue of classical polynomial resultants in the following sense: if a (rigidity) circuit \(C\) is obtained as the combinatorial resultant \(\operatorname{CRes}(A, B, e)\) of two circuits \(A\) and \(B\) with the edge \(e\) eliminated, then the resultant \(\operatorname{Res}(p_{A}, p_{B}, x_e)\) of circuit polynomials \(p_{A}\) and \(p_{B}\) with respect to the indeterminate \(x_e\) is supported on \(C\) and contained in the elimination ideal \( \langle p_C \rangle\) generated by the circuit polynomial \(p_C\) . When \(\operatorname{Res}(p_{A}, p_{B}, x_e)\) is irreducible then it will be equal to \(p_C\) . However, in general \(p_C\) will only be one of its irreducible factors over \(\mathbb Q\) . In fact exactly one factor (counted with multiplicity) of \(\operatorname{Res}(p_{A}, p_{B}, x_e)\) may correspond to \(p_C\) , and that factor can be deduced by examining the supports of the factors and performing an ideal membership test on those factors that have the support of \(p_C\) .

These facts are summarized by Algorithm 8.1, where the work to clean up the resultant in order to extract the circuit polynomial is presented as the separate Algorithm 8.2. The rest of this section is devoted to the proof of correctness of Algorithms 8.1 and 8.2, along with several remaining open problems.

We now have all the ingredients to describe an algorithmic solution to the main problem stated in the introduction: given a rigidity circuit \(C\) , compute its circuit polynomial \(p_C\) .

One way of doing this is captured by Algorithm 9.1. It uses a combinatorial circuit-resultant (CCR) tree \(T_C\) that was precomputed with Algorithm 3.1. It inductively computes polynomials supported by circuits at levels of the tree closer to the root from polynomials supported on circuits on a higher level. This algorithm stores all circuit polynomials on one level prior to going to the next level. The method becomes impractical when the CCR tree has a large number of vertices on some level, as would be the case, say, when the binary CCR tree is balanced. The correctness of Algorithm 9.1 follows directly from Algorithms 3.1 and 8.1.

Algorithm 9.2 takes an alternative approach and traverses the CCR tree in postfix order. This is naturally described as a recursive procedure. The recursion stack retains left child circuit polynomials along a path to a node from the root in the CCR tree, and thus its space complexity depends on the depth of the tree.

We generalize the algorithms in section 9 by allowing all dependent sets in the rigidity matroid at the nodes, with the aim of improving computational performance.

First we relax some of the constraints imposed on the resultant tree by the construction from subsection 3.4. The internal nodes correspond, as before, to combinatorial resultant operations, but (a) they are no longer restricted to be applied only on circuits or to produce only circuits, (b) the leaves can be labeled by graphs other than \(K_4\) ’s, and (c) the sequence of graphs on the nodes along a path from a leaf to the root is no longer restricted to be strictly monotonically increasing in terms of the graphs’ vertex sets.

The generators in \(\operatorname{Gen}\) will be the graphs allowed to label the leaves. For the purpose of generating (combinatorial) circuits and computing (algebraic) circuit polynomials, we choose a set of generators, discussed in section 11, that are dependent in the rigidity matroid.

Hence, CCR trees are special cases of CR trees. An example of a CR tree which is not a CCR tree is illustrated in Figure 12.

The example in Figure 12 is a valid combinatorial resultant tree for the \(K_{3,3}\) -plus-one circuit. After reviewing the necessary algebraic notions in the next section, we will use it in subsection 13.4 to demonstrate our generalized algebraic elimination algorithm described in section 12.

We work with the set \(\operatorname{GenCM}_{n}\) of generators for the 2D Cayley–Menger ideal \(\operatorname{CM}_{n}\) as given by the set of all \(5\times 5\) minors of the \((n+1)\times (n+1)\) Cayley matrix. Each generator \(g\in \operatorname{GenCM}_{n}\) is identified with its support graph \(G_g\) , as defined in section 7. To motivate the possible choices for the family of graphs \(\operatorname{Gen}\) for the generalized combinatorial resultant trees defined in section 10, we now tabulate the support graphs of all generators, up to multiplication by a nonzero constant, relabeling, and graph isomorphism.

To find all these graphs, it is sufficient to consider the set \(\operatorname{GenCM}_{10}\) of all \(5\times 5\) minors of \(\operatorname{CM}_{10}\) . Using a computer algebra package, we can verify that this set has 109 619 distinct minors, of which 106 637 have distinct support graphs. The IsomorphicGraphQ function of Mathematica was used to reduce them to the 14 graph isomorphism classes, 11 of which are shown in Figure 13. The only two representatives with fewer than 6 vertices are \(K_4\) and \(K_5\) . There are three isomorphism classes on 6, 7, 8 vertices (one is \(K_6\) ), two on 9, and one on 10 vertices. The corresponding generator polynomials are, up to isomorphism (relabeling of variables induced by relabeling of the vertices), unique for the given support, with a few exceptions: for \(K_5\) , we found 3 distinct (nonisomorphic) polynomials.

Note that there may be polynomials in \(\operatorname{CM}_{n}\) supported on the same set as a generator from \(\operatorname{GenCM}_{n}\) , but which themselves do not arise from a single \(5\times 5\) minor of a Cayley matrix. For example, if \(p\in \operatorname{GenCM}_{n}\) is supported on a \(K_5\) and \(q\in \operatorname{GenCM}_{n}\) is supported on a \(K_6\) such that \(\operatorname{supp}{p} \subset \operatorname{supp}{q}\) , then \(p+q\) has the support of a generator on \(K_6\) but itself is not in \(\operatorname{GenCM}_{n}\) .

We now have all the ingredients for describing Algorithm 12.1, which computes the circuit polynomial \(p_C\) for a circuit \(C\) from a given combinatorial resultant tree \(T_C\) , or returns a message that \(p_C\) cannot be computed using \(T_C\) . Just like the algorithms of section 9, it computes resultants at each node of the tree, starting with the resultants of generators of \(\operatorname{CM}_{n}\) supported on leaf nodes. At the root node the circuit polynomial for \(C\) is extracted from the irreducible factors of the resultant at the root. The main difference lies at the intermediate (nonroot) nodes, as described in Algorithm 12.1 below. This is because the polynomials sought at nonleaf nodes, not being supported on circuits, are not necessarily irreducible polynomials supported on the desired dependent graph, as was the case in section 9. Hence, conceivably, they may have factors that are not in the Cayley–Menger ideal, and it might be the case that none of their factors that are in the Cayley–Menger ideal are supported on the desired graph, but their product with other factors is. Moreover, it might be the case that an intermediate resultant \(\operatorname{Res}(f, g, x)\) is zero, with \(x\) being present only in the supports of common factors of \(f\) and \(g\) , in which case the algorithm cannot resume along the chosen tree \(T_C\) . It remains, however, as an open question (which may entail experimentation with gigantic polynomials) to explicitly find such examples (we did not find any so far) and to prove what may or may not happen.

Proof of correctness of Algorithm 12.1. Recall that \(\mathbb{Q}[G]\) denotes the ring of polynomials with indeterminates \(x_{ij}\) with \(i \lt j\) given by the edges \(ij\) of \(G\) .

Steps 1–2. Compute the resultant. If the resultant is zero, the algorithm terminates with the message that it is not possible to continue along \(T_C\) . We can attempt to replace one or both of \(v\) and \(w\) with other polynomials in \(\operatorname{CM}_{n}\) with appropriate support that would lead to a nonzero resultant; however, in our presentation we assume that all the choices made in previous calls of Algorithm 12.1 (e.g., the choice of a candidate in line 9) remain fixed.

Step 3. The elimination ideal \(\operatorname{CM}_{n} \cap \mathbb{Q}[G]\) is prime, and hence at least one irreducible factor \(p\) of \(r\) is in \(\operatorname{CM}_{n}\) .

Step 4. If there is exactly one factor \(p\) supported on a dependent set, then that factor must necessarily be in \(\operatorname{CM}_{n}\) . This follows from the primality of \(\operatorname{CM}_{n} \cap \mathbb{Q}[G]\) : assume for simplicity that \(r\) factors as \(q_1\cdot q_2\cdot p\) , with only \(p\) being supported on a dependent set. If \(q=q_1\cdot q_2\) is supported on an independent set, then it is not in \(\operatorname{CM}_{n}\) , and hence \(p\) must be in \(\operatorname{CM}_{n} \cap \mathbb{Q}[G]\) . If \(q\) is supported on a dependent set, then \(q\in \operatorname{CM}_{n}\) would imply that one of \(q_1\) or \(q_2\) is in \(\operatorname{CM}_{n}\) , but none of the two are. Therefore \(p\in \operatorname{CM}_{n} \cap \mathbb{Q}[G]\) in any case.

Steps 5–6. There are now two possibilities for \(p\) : either it is supported on \(G\) , in which case we return it, or it is supported on a proper subset of \(G\) . If its support is a proper subset of \(G\) , we can in principle return any polynomial \(q p\) such that \(\operatorname{supp}{q p}=G\) . Recall that the resultant is multiplicative (Proposition 21); hence in a subsequent invocation of the algorithm, in the computation of \(\operatorname{Res}(qp, f, y)=\operatorname{Res}(q, f, y)\operatorname{Res}(p, f, y)\) for some \(f\) and \(y\) we can keep the factor \(\operatorname{Res}(q, f, y)\) unevaluated. An alternative would be to modify the resultant tree \(T_C\) by replacing \(G\) with the graph \(G_p\) given by the support of \(p\) (as defined in section 7). However, in our presentation we keep the resultant tree fixed throughout and choose \(q\) to simply be the product \(\Pi_{x\in G\!\setminus\! \operatorname{supp}{p}}x\) of all indeterminates in \(G\!\setminus\! \operatorname{supp}{p}\) .

In our experiments we are yet to encounter an example in which an irreducible factor supported on a dependent set that is a proper subset of \(G\) appears. We leave as an open problem to find such an example, or prove that it cannot occur.

Steps 7–12. If there is more than one irreducible factor supported on a dependent set, we store them in the list candidates in some order. Factors are then tested for membership in \(\operatorname{CM}_{n}\) with an ideal membership test, in the order in which they are stored in the list candidates. The first irreducible factor that passes the test is returned if its support is \(G\) , or it is completed to a polynomial supported on \(G\) in the same way as described above and then returned.

We have not encountered examples in which more than one irreducible factor supported on a dependent set appeared; however, this is most likely because we were only able to perform computations on graphs with up to 8 vertices.

Since \(G\) is not necessarily a circuit, the elimination ideal \(\operatorname{CM}_{n} \cap \mathbb{Q}[G]\) is no longer necessarily principal, and we can no longer guarantee the existence of a unique irreducible factor \(p\) of \(r\) that is supported both on \(G\) and in \(\operatorname{CM}_{n}\) . We have not encountered this possibility in our experiments, and we leave it as an open question.

Refinements of Algorithm 12.1. If at a node of \(T_C\) we have \(\operatorname{Res}(v, w, e)=0\) , we can attempt to replace \(v\) or \(w\) with other appropriate polynomials in \(\operatorname{CM}_{n}\) . In particular we can attempt to recompute \(v\) or \(w\) by choosing a different polynomial from the list of candidates in line 9. This approach, however, might require recomputing \(v\) and \(w\) many times, and we can still not guarantee that \(\operatorname{Res}(v, w, x_e)\) would be nonzero. We leave as an open problem to find the conditions on \(v\) and \(w\) so that \(\operatorname{Res}(v, w, x_e)\) is not zero.

Alternatively we can replace one or both branches of the resultant tree for \(G\) (taken as the subtree of \(T_C\) rooted at \(G\) ) with a tree that would lead to a nonzero resultant at \(G\) . For that purpose it would be useful to have an algorithm that enumerates the resultant trees of a dependent graph. Such enumeration appears to be much more challenging than for CCR trees (Open Problem 13), and it is unclear that an efficient solution to the following problem can be obtained.

If the answer to Open Problem 42 is positive, we have to decide which polynomial to output. In Algorithm 12.1 the first irreducible factor with dependent support that passes the ideal membership test is chosen and returned (possibly padded by the indeterminates in \(G\!\setminus\! \operatorname{supp}{p}\) ). However, it may be the case that the first irreducible factor that passes the ideal membership test is not the best choice if what we have in mind is the goal of simplifying the resultant computation when this algorithm is invoked on the parent of \(G\) . For example, relative to the remaining factors that pass the ideal membership test, the first factor that passed the test could have a very large degree in the indeterminate that is to be eliminated in the subsequent invocation of the algorithm, which, as a consequence, would lead to a very large dimension of the Sylvester determinant.

We propose the following decision criteria in the case when \(r\) has multiple irreducible factors \(\{p_1,\dots,p_k\}\) in \(\operatorname{CM}_{n}\) . From the set \(\{p_1,\dots,p_k\}\) choose the polynomial:Criterion (i) ensures that when the algorithm is invoked on the parent of \(G\) , the dimension of the Sylvester determinant will be the least possible; criterion (ii) ensures that the resultant will be of least possible homogeneous degree (Proposition 23), while criterion (iii) minimizes the total number of monomials that appear as entries in the Sylvester determinant.

This choice of decision criteria may not be the best possible, and we leave as an open problem to formulate other decision criteria.

In this section we discuss our experimental work, carried out with the algorithms presented in this paper, that led to effective computations of all circuit polynomials in \(\operatorname{CM}_{6}\) . Table 1 summarizes the results. To the best of our knowledge, except for the circuit polynomial of \(K_4\) , these polynomials have not been computed before. Each example of a circuit polynomial is presented up to relabeling of vertices. All the circuit polynomials computed in this section are available at the GitHub repository [39]. For comparison purposes, we also include some preliminary calculations done or attempted with Gröbner basis methods.

The \( {K_4}\) circuit. The only circuit polynomial that is directly obtainable as a generator of \(\operatorname{CM}_{n}\) for any \(n\geq 4\) , and does not require Gröbner basis methods or resultant computations, is the circuit polynomial of a \(K_4\) graph (possibly relabeled). This polynomial has 22 terms, homogeneous degree 3, and is of degree 2 in any of its variables.

In this paper we introduced the combinatorial resultant operation, analogous to the classical resultant of polynomials. We offer here some final comments and suggestions for further research.

Irreducibility test. Our methods still have several computational drawbacks, in that they require irreducibility checks, with a possible further factorization and an ideal membership test for those factors that have the support of a circuit.

Ideally we would like to detect combinatorially when a resultant of two circuit polynomials that has the support of a circuit will be irreducible. The absolute irreducibility test of Gao [20], which states that a polynomial is absolutely irreducible if and only if its Newton polytope is integrally indecomposable, in conjunction with the description of the Newton polytope of the resultant of two polynomials by Gelfand, Kapranov, and Zelevinsky [21, 22], gives a combinatorial criterion for absolute irreducibility, but not for irreducibility over \(\mathbb Q\) . However, not every circuit polynomial is absolutely irreducible; for example, the circuit polynomial of a wheel on 4 vertices is irreducible over \(\mathbb Q\) but not absolutely irreducible.

What we observed in practice. It is worth noticing that whenever in our computations we had to decide which factor of a resultant belonged to \(\operatorname{CM}_{n}\) , we never had to perform an ideal membership test. It was always sufficient to inspect only the supports of the irreducible factors of the resultant. In all cases where the calculation succeeded, all but one irreducible factor were supported on Laman graphs, and one factor was supported on a dependent set. It seems unlikely that this is the general case, and it would be of interest to determine under which conditions the resultant has exactly one factor (up to multiplicity) supported on a dependent set in \(\operatorname{\mathbf{\mathcal A}}(\operatorname{CM}_{n})\) .

Open problems. We conclude the paper with a few more open problems concerning the algebraic and geometric structure of the resultant of two circuit polynomials.

This question appears to be very challenging. The answer depends heavily on the specific polynomials \(p_A\) , \(p_B\) and the variable \(x_e\) and pertains to the relationship between (affine) varieties related to \(r=\operatorname{Res}(p_A, p_B, x_e)\) , \(p_A\) , and \(p_B\) . Let \(R=\mathbb C[C]\) be a polynomial ring, let \(p_A,p_B\in R[x_e]\) , and let \(I_{x_e}\) denote the elimination ideal \( \langle p_A,p_B \rangle \cap R\) . Let \(l_A\) (resp., \(l_B\) ) be the leading coefficient of \(p_A\) (resp., \(p_B\) ) with respect to \(x_e\) . Then by the Extension Theorem [13, Theorem 8 in section 6 of Chapter 3] and the Closure Theorem [13, Theorem 4 in section 4 of Chapter 4] we have the following equality of (affine) varieties: \(V(r)=V(l_A,l_B)\cup V(I_{x_e})\) . Furthermore, if \(r\) factors as \(q\cdot p_C^k\) for some positive integer \(k\) , then \(V(r)=V(q)\cup V(p_C)\) . Ideally we would want \(V(r)=V(p_C)=V(I_{x_e})\) but in general \(V(p_C)\) is only contained in \(V(I_{x_e})\) and \(V(r)\) . Hence the structure of \(V(r)\) , in particular its irreducibility, depends on algebraic data \(V(l_A,l_B)\) and \(V(I_{x_e})\) , whose relationship to the combinatorial, graph-theoretical data is yet to be found.

Further interesting questions pertain to parameters of circuit polynomials such as the degree in a single variable or the number of monomials. The first one, the degree with respect to a single variable \(x_e\) in the support of a circuit polynomial, is related to the literature on the number of embeddings of Laman graphs, where the best known upper bound is \(3.77^n\) [2] for \(n\) vertices. Bounds on the degree of an individual indeterminate of a 3-connected circuit polynomial can be inferred from here, while for the 2-connected ones their decomposition into 3-connected components is needed. On the other hand, we are not aware of any such bounds on the number of monomial terms of circuit polynomials, but have observed that their number quickly becomes large, as shown in the Table 1.

We would like to thank the anonymous reviewers for their comments, which have helped improve the presentation, and for suggesting references that have increased the scope of the paper.