Polynomial time algorithms in invariant theory for torus actions
Peter Bürgisser, M. Levent Do?an, Visu Makam, Michael Walter, Avi Wigderson
aa r X i v : . [ c s . D S ] F e b POLYNOMIAL TIME ALGORITHMS IN INVARIANT THEORYFOR TORUS ACTIONS
PETER B ¨URGISSER, M. LEVENT DO ˘GAN, VISU MAKAM, MICHAEL WALTER, AND AVI WIGDERSON
Abstract.
An action of a group on a vector space partitions the latter into a set of orbits. Weconsider three natural and useful algorithmic “isomorphism” or “classification” problems, namely, orbit equality , orbit closure intersection , and orbit closure containment . These capture and relate toa variety of problems within mathematics, physics and computer science, optimization and statistics.These orbit problems extend the more basic null cone problem, whose algorithmic complexity hasseen significant progress in recent years.In this paper, we initiate a study of these problems by focusing on the actions of commutativegroups (namely, tori). We explain how this setting is motivated from questions in algebraic com-plexity, and is still rich enough to capture interesting combinatorial algorithmic problems. Whilethe structural theory of commutative actions is well understood, no general efficient algorithmswere known for the aforementioned problems. Our main results are polynomial time algorithmsfor all three problems. We also show how to efficiently find separating invariants for orbits, andhow to compute systems of generating rational invariants for these actions (in contrast, for poly-nomial invariants the latter is known to be hard). Our techniques are based on a combination offundamental results in invariant theory, linear programming, and algorithmic lattice theory. Contents
1. Introduction 21.1. Algorithms in invariant theory 21.2. Orbit problems 31.3. Torus actions and main results 51.4. Further motivation and algorithmic applications 81.5. Organization of the paper 92. Preliminaries of invariant theory 103. Invariants and orbit closures of torus actions 113.1. Representations and invariants 123.2. Newton cone and orbit closures 134. Generating Laurent polynomials and rational invariants 154.1. Invariant Laurent polynomials 164.2. Rational invariants 175. Orbit equality problem 185.1. Laurent monomial equivalence 186. Orbit closure intersection and explicit separating invariants 196.1. Reduction to orbit equality 206.2. Explicit separating invariant 207. Orbit closure containment 218. Orbit problems for compact tori 229. Concluding remarks, future directions, and open problems 23Acknowledgements 24References 24 . Introduction
Consider the following two problems, which on the face of it have nothing to do with each other:(1) Will the cue ball’s trajectory on a billiards table ever end up in a pocket?(2) Given a bipartite graph G , and two functions w , w ′ assigning weights to edges, is it thecase that they assign the same weight to every perfect matching M of G ?Both turn out to be orbit problems for torus actions, and exemplify the class of problems we studyin this paper.As our introduction is somewhat long, we break it up as follows. We start with general back-ground to algorithmic invariant theory in § § § § § Algorithms in invariant theory.
Computational invariant theory is a subject whose originscan be traced back to “masters of computation” in the 19th century such as Boole, Gordan,Sylvester and Cayley among others. The second half of the 20th century injected a major impetusto both structural and computational aspects of these mathematical areas. On the one hand,the advent of digital computers allowed mathematicians means to study much larger such algebraicstructures than could be accessed by hand. On the other, the parallel development of computationalcomplexity provided a mathematical theory with precise computational models for algorithms andtheir efficiency analysis. This combination has injected many new ideas and questions into invarianttheory and related fields, leading to the development of algorithmic techniques such as Gr¨obnerbases and many others, which supported faster and faster algorithms. Texts on this large body ofwork can be found, for example, in the books [DK15, Stu08, CLO97]. While the computationalcomplexity put focus on polynomial time as the staple of efficiency, it also provided means to arguethe likely impossibility of such fast algorithms for certain tasks, through the Cook-Karp-Levintheory [Coo71, Kar72, Lev73] of NP-completeness (for Boolean computation) and Valiant’s theoryof VNP-completeness.More recently, a further surge in collaboration between algebraists and complexity theorists onthese algorithmic questions in invariant theory and representation theory arose from two (related)sources starting in the turn of this century. Both imply that these very algorithmic questions inalgebra are deeply entwined with the core complexity questions of P vs. NP and VP vs. VNP. Notsurprisingly, new enriching connections between these two research directions are newly found asthey develop, providing an exciting collaboration.The first source is Mulmuley and Sohoni’s Geometric Complexity Theory (GCT) [MS01], whichhighlights the inherent symmetries of complete problems of these complexity classes, and throughthese suggests concrete invariant theoretic and representation theoretic attacks on the questionsabove. This has lead to many new questions, techniques, and much faster algorithms (see, forexample, [Mul17, FS13, BCMW17, MW19]).The second source is the work of Impagliazzo and Kabanets [KI04], using Valiant’s completenesstheory for VP and VNP to again attack these major complexity problems directly through thedevelopment of efficient deterministic algorithms for the basic PIT (Polynomial Identity Testing)problem. This problem, which (again, thanks to Valiant’s completeness) has natural symmetries, isvery similar to basic invariant theory problems. Major progress was recently made on resolving suchrelated algorithmic problems, starting with [Gur04a, GGOW16, IQS17, IQS18, DM17b]. Manyothers continue to follow, see, for example, [DM20a, AZGL +
18, GGOW20, BGO +
18, BFG + + +
19] for a recent description of the state-of-art. .2. Orbit problems.
We now briefly describe the basic setting and problems of interest, postpon-ing some of the technical details to later sections for the sake of brevity. A group homomorphism ρ : G → GL( V ), where V is a vector space (always complex and finite-dimensional) is called a rep-resentation of G . One can think of this as a (linear) action of G on V , i.e., a map G × V → V where( g, v ) ρ ( g ) v satisfies the usual axioms of a group action. For us, groups will always be algebraicand representations rational, that is, morphisms of algebraic groups. We will denote ρ ( g ) v by gv or g · v .For v ∈ V , we define its orbit O v := { gv | g ∈ G } (denoted O G,v if the group is not clear fromcontext) to be the subset of points that can be reached from v by applying a group element. Wedenote by O v the topological closure of O v . These notions are extremely basic and in many concreteinstances very familiar. One simple example is the action of GL n × GL n on n × n matrices by leftand right multiplication: clearly, the orbit of a matrix A consists of the matrices having the samerank as A ; moreover, the orbit closure of A is the set of matrices whose rank is at most the rankof A . Another example is the conjugation action of GL n on n × n matrices, where the orbits arecharacterized by Jordan normal forms. Understanding the space of orbits of a given group action is perhaps the most basic task ofinvariant theory. The following three basic algorithmic problems will be the focus of this paper.
Problem 1.1.
Let ρ : G → GL( V ) be a representation of a group G . Given v, w ∈ V :(1) Orbit equality:
Decide if O v = O w ;(2) Orbit closure intersection:
Decide if O v ∩ O w = ∅ ;(3) Orbit closure containment:
Decide if w ∈ O v . As we will discuss the computational complexity of algorithms for these problems, one needs tospecify how inputs are given and how we measure their size. We will discuss this, but for now itsuffices to think of n = dim( V ), the degree of ρ (assuming it is a polynomial function), and thebit-length of the input vectors v, w as the key size parameters.The aforementioned problems capture and are related to a natural class of “isomorphism” or“classification” problems across many domains in mathematics, physics and computer science.Examples include the graph isomorphism problem [Der13], non-commutative rational identity test-ing [GGOW16, IQS18], equivalence problems on quiver representations [DM17a, DM18], matrixand tensor scaling [BGO +
18, BFG + +
93] and moduleisomorphism problems [BL08].To briefly hint at the role of invariant theory, let us take a closer look at problem (2), that is, theproblem of orbit closure intersection. We denote by C [ V ] the ring of polynomial functions on V . Apolynomial function f on V is called invariant if it is constant along orbits, i.e., f ( gv ) = f ( v ) forall g ∈ G and v ∈ V . The collection of all invariant polynomials forms a subring C [ V ] G , called the invariant ring . Since polynomials are continuous, invariant polynomials are constant along orbitclosures. In particular, two points v and w are indistinguishable by invariant polynomials whentheir orbit closures intersect. Amazingly, the converse is also true for a large class of group actionsthanks to a result due to Mumford: if the orbit closures of v and w do not intersect, then they canalways be distinguished by an invariant polynomial. See Theorem 2.1 for a precise statement.Mumford’s theorem suggests an approach to orbit closure intersection – test if f ( v ) = f ( w ) forall invariant polynomials f . For this strategy to be effective, one needs a computational handleon invariant polynomials. Naively there are infinitely many polynomials, but a foundational result The orbit closures of two matrices intersect if and only if the matrices have the same eigenvalues (counted withmultiplicity). The special case of w = 0 is called the null cone membership problem. In fact, many of the recent algorith-mic advances mentioned above efficiently solve the null cone problem for specific group actions, see [BFG +
19] andreferences therein. The motivation of this paper is to extend that understanding to these more general problems. f Hilbert helps tackle this issue. A system of generating polynomial invariants is a collection ofinvariant polynomials { f , . . . , f r } such that any other invariant polynomial can be written as apolynomial in the f i ’s. In particular, to test for orbit closure intersection it suffices to test whethereach of the f i take the same value on both points. Hilbert showed the existence of a finite system ofgenerating polynomial invariants and also gave an algorithm to produce them [Hil90]. Since then,many improvements on the complexity of such algorithms were developed, but even today this taskis, in general, infeasible. One basic obstacle is the very description of such a system of generatinginvariants, coming both from the size of this set and the degree of each polynomial in it.Nearly a century later, a (singly) exponential bound (in n ) on the degrees of a system of gener-ating polynomial invariants was achieved for a very general class of group actions [Der01], whichis unfortunately the best possible in this generality, see [DM20b]. A singly exponential bound isnecessary to capture a polynomial with a poly-sized (in n ) arithmetic circuit, but is by no meanssufficient. Another issue that one has to deal with is the number of invariants in a system ofgenerating polynomial invariants, and it is often the case that there are exponentially many in anysystem. This led Mulmuley [Mul17] to suggest the notion of a succinct circuit as a way to capturea system of generating polynomial invariants with a view towards using them for orbit closureintersection. Unfortunately, this approach does not seem to be computationally feasible either.See [GIM +
20] where Mulmuley’s conjecture [Mul17, Conjecture 5.3] on the existence of succinctcircuits was disproved under natural complexity assumptions. What is perhaps most surprising isthat this already happens for a commutative group action, namely when G is a torus. Further, anexample of a group action was given where any system of generating polynomial invariants mustcontain a VNP-hard polynomial.The negative result above seem to suggest that the algorithmic tasks at hand are infeasible, evenfor torus actions, i.e., groups of the form ( C × ) d . The main results of our paper show the opposite: all of them are efficiently solvable for torus actions! The main novelty on our approach is using rational invariants instead of polynomial invariants.A rational invariant is a quotient of polynomials that is invariant, see Section 1.3 for a precisedefinition. This is a bit unexpected since Mumford’s theorem simply does not extend to rationalinvariants: it is easy to construct examples where two points whose orbit closures intersect aredistinguished by a rational invariant. Yet, for representations of tori, we show that (a certain specialcollection of) rational invariants can be used (in a delicate way) to capture not just orbit closureintersection, but orbit closure containment and orbit equality as well. Moreover, we show thatrational invariants are computationally easy in this case, in stark contrast with the aforementionedhardness results for polynomial invariants [GIM + +
20] along with our results on rational invariants suggestthat a more thorough investigation of rational invariants is needed in the case where the actinggroup is non-commutative, e.g., SL n . For example, the permanent of an n × n matrix, which has degree n , is believed to require exponential circuitsize. This is essentially the content of Valiant’s proof that the permanent is complete for the class VNP, combinedwith the hypothesis that VNP = VP. This is already the case for the matrix scaling action discussed in Section 1.4. .3. Torus actions and main results.
We now discuss the main contributions of our paper inmore detail and precision. Our results concern torus actions, so we specialize the discussion of thepreceding section and consider a d -dimensional complex torus T = ( C × ) d as the acting group G .The group law is just pointwise multiplication, i.e., ( t , . . . , t d ) · ( s , . . . , s d ) = ( t s , . . . , t d s d ).Any linear action of a torus can be described by an integer matrix M ∈ Mat d,n ( Z ) called the weight matrix (where Mat d,n ( Z ) denotes the space of d × n integer matrices). The representa-tion ρ M : T → GL n ( C ) corresponding to a weight matrix M = ( m ij ) looks as follows: ρ M ( t ) = Q di =1 t mi i ... Q di =1 t mini (1.1)Thus any torus action can be viewed as a scaling action, where each coordinate is scaled separatelyaccording to a Laurent monomial. The weight matrix (up to reordering of columns) determines therepresentation. Despite the simple description of commutative torus actions, they as well capturefundamental notions, and the associated orbits can be quite complex. One example is the matrixscaling problem, where the orbits capture weights of perfect matchings (see Problem 1.7).In this paper, we will assume that a torus action is given by specifying the weight matrix. Thusthe bit-length of the entries of the weight matrix are included in the input size of the problems.Moreover, we will allow complex number inputs. These can be described up to finite precision byelements in the field of Gaussian rationals Q ( i ) = { s + it | s, t ∈ Q } , which will be encoded in thestandard way; see, e.g., [Mul17]. The following theorem captures the main results of our paper.
Theorem 1.2.
Given as input a weight matrix M ∈ Mat d,n ( Z ) as well as vectors v, w ∈ Q ( i ) n ,denote by b the maximal bit-length of the entries of v, w , and M . Then we can in time poly( d, n, b ) :(1) decide whether O v = O w ;(2) decide whether O v ∩ O w = ∅ ;(3) decide whether w ∈ O v .In other words, for rational representations of tori, there are polynomial time algorithms for orbitequality, orbit closure intersection, and orbit closure containment. We note that the null cone membership problem mentioned earlier, namely Problems 1.1 (2)/(3)when the input vector w is the 0 vector, was known to have a polynomial time algorithm by asimple reduction to linear programming. There is no known way of doing the same for the orbitproblems above, and indeed our theorem above takes an alternative route.While one might hope for efficient algorithms for Problems 1.1 (1) and (2) in much more generalsituations than for tori (for general reductive group actions), our efficient algorithm for orbit closurecontainment is in stark contrast to the known NP-hardness of the general orbit closure containmentproblem [BIL + We can also describe this action as follows: Identify v ∈ C n with a Laurent polynomial P nj =1 v j z m j · · · z m dj d ;then the action of T corresponds precisely to rescaling the variables z , . . . , z d [Gur04b]. In fact, our results hold more generally when the elements in Q ( i ) are given in a ‘floating point’ format, namelyin the form ( s + it )2 p , with s, t ∈ Q and p ∈ Z encoded in binary in the standard way. The same is true for input ofthe form 2 p , with p ∈ Q encoded in binary. See Remark 5.6. Namely, a vector v is in the null cone if and only if the convex hull of the weights corresponding to the nonzerocoordinates of v does not contain the origin. an be quite complicated. However, suppose that we restrict to vectors of some fixed support, i.e.,“nonzero pattern” of the coordinates. This restriction is without loss of generality, since twovectors can only be in the same orbit when their supports coincide. However, it allows us to studya richer class of functions, namely Laurent polynomials instead of ordinary polynomials. Allowingfor negative exponents makes an important difference: while polynomial invariants naturally forma semigroup, invariant Laurent polynomials form a lattice , isomorphic to the integral vectors in thekernel of the weight matrix. Lattices are much better behaved than semigroups, for example theyhave small bases which can be found efficiently.Before describing our results, let us define invariant Laurent polynomials more precisely. For arepresentation ρ : G → GL( V ) of a group G , we have an action of G on the polynomial ring C [ V ]defined by ( g · f )( v ) := f ( ρ ( g ) − v ). When V = C n , we can identify C [ V ] = C [ x , . . . , x n ] with thepolynomial ring in n variables. Now consider the set of vectors with nonzero coordinates in S ⊆ [ n ]: X S = { v ∈ C n | v j = 0 if and only if j ∈ S } . The Laurent polynomials in the variables x j for j ∈ S form the natural class of functions on X S (since we can always divide by the nonzero coordinates). Accordingly, we will denote their collectionby C [ X S ]. Now, for a torus action of the form (1.1), the group T acts on any monomial x c = x c · · · x c n n by a simple rescaling. Accordingly, we also have an action of T on the algebra ofLaurent polynomials C [ X S ]. A Laurent polynomial f is called invariant if g · f = f for all g ∈ G .Clearly, if f is invariant, then so are all the Laurent monomials occuring in f . The collection ofall invariant Laurent polynomials forms the subalgebra C [ X S ] G of invariant Laurent polynomials .A collection of invariant Laurent polynomials f , . . . , f r is called a system of generating invariantLaurent polynomials in the variables { x j } j ∈ S if they generate C [ X S ] G as an algebra. For torusactions, these can always be taken to be Laurent monomials , in which case we call them a systemof generating invariant Laurent monomials . We can then state our key result: Theorem 1.3.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d , andlet S ⊆ [ n ] . Assume that the bit-lengths of the entries of M are bounded by b . Then, in poly( d, n, b ) -time, we can construct an arithmetic circuit with division C whose output gates compute a systemof generating invariant Laurent monomials f , . . . , f r in the variables { x j } j ∈ S , where r ≤ n . Here we recall the notion of an arithmetic circuit with division , which is a directed acyclic graphas follows. Every node of indegree zero is called an input gate and is labeled by either a variableor a rational (complex) number. Nodes of indegree one and outdegree one are labeled by − andare called divison gates. Nodes of indegree two and outdegree one and is labeled either + or × ;in the first case it is a sum gate and in the second a product gate. The only other nodes allowedare output gates which have indegree one and outdegree zero. Given an arithmetic circuit withdivision, it computes a rational function at each output node in the obvious way. The bit size ofsuch an arithmetic circuit is the total number of nodes plus the total bit-length of the specificationof all rational numbers computed in all of its gates. The notion of (division free) arithmetic circuits is obtained by disallowing division gates. They compute polynomials in the obvious way.We emphasize that the number of generators produced by Theorem 1.3 is at most n (in partic-ular, independent of the bit-length b ), in stark contrast to the situation for monomial invariants.Moreover, the bit-length of C is polynomially bounded.As a consequence of Theorem 1.3, we are also able to construct arithmetic circuits that computea generating set of rational invariants . For a representation ρ : G → GL( V ), the action of G onthe polynomial ring C [ V ] always extends to an action on its field of rational functions, the rationalfunctions C ( V ). A rational function f ∈ C ( V ) is called invariant if g · f = f for all g ∈ G . Thecollection of all rational invariants forms the sub-field C ( V ) G of rational invariants . A collection In the language of algebraic geometry, these are the “regular” functions on X S . f rational invariants f , . . . , f r ∈ C ( V ) is called a system of generating rational invariants if theygenerate C ( V ) G as a field extension of C . Note that any invariant Laurent polynomial is a rationalinvariant, but the converse is not necessarily true. Nevertheless: Corollary 1.4.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d . Assumethat the bit-lengths of the entries of M are bounded by b . Then, in poly( d, n, b ) -time, we canconstruct an arithmetic circuit with division C whose output gates compute a system of generatingrational invariants f , . . . , f r ∈ C ( x , . . . , x n ) T , where r ≤ n . This result is in distinct contrast to the impossibility of finding succinct circuits for generating polynomial invariants under natural complexity assumptions [GIM + O v ∩ O w = ∅ , then we can construct in polynomial time an arithmetic circuit computinga separating invariant monomial that can serve as a “witness” of the non-intersection. Corollary 1.5.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d . Let v, w ∈ Q ( i ) be such that O v ∩ O w = ∅ . Assume the bit-lengths of the entries of v, w and M are bounded by b . Then, in poly( d, n, b ) -time, we can construct an arithmetic circuit of bit-length poly( d, n, b ) , which computes an invariant monomial f such that f ( v ) = f ( w ) . So far, we have discussed orbit problems for complex tori T = ( C × ) d . It is interesting to ask towhich extent our results hold for “compact” tori, which are groups of the form K = (S ) d , whereS = { z ∈ C × | | z | = 1 } . Besides the fundamental algorithmic interest in this setting, such groupactions are important in several areas. For example, the time evolution of periodic systems inHamiltonian mechanics are naturally given by S -actions, and important symmetries in classicaland quantum physics are given by compact group actions.In fact, the results discussed so far can also be used to give an efficient solution for orbit problemsfor compact tori. Any (continuous) finite-dimensional representation of (S ) d extends to a repre-sentation of ( C × ) d , so representations are specified as before by a weight matrix M ∈ Mat d,n ( Z ).Moreover, the compactness implies that orbits O K,v = { kv | k ∈ K } are closed and so all threeproblems mentioned in Problem 1.1 coincide. Therefore, the following corollary solves all threeproblems for compact tori: Corollary 1.6.
Let the weight matrix M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d and put K = (S ) d . Further, let v, w ∈ Q ( i ) n and assume that the bit-lengths of theentries of v, w and M are bounded by b . Then, in poly( d, n, b ) -time, we can decide if O K,v = O K,w . To give additional context to this result, we briefly mention some recent results achieving poly-nomial time algorithms for orbit closure intersection of specific group actions. For the left-rightaction (of SL n × SL n on m -tuples of n × n matrices), one approach to solving the orbit closureintersection problem is to (approximately) reduce to the orbit equality problem for the maximalcompact subgroup (which happens to be SU( n ) × SU( n ), where SU( n ) denotes the group of n × n unitary matrices with determinant 1), see[AZGL + +
19] and referencestherein), it is natural to ask if a similar approach could be useful for general reductive group actions.For torus actions, interestingly, we can also go in the other direction. Namely, our result for theorbit equality problem for the maximal compact subgroup, Corollary 1.6, is derived from our mainresult for complex tori, i.e., Theorem 1.2. More generally, we observe that for arbitrary reductivegroup actions, the orbit equality problem for the maximal compact subgroup is always equivalent Note that K is indeed compact, and a subgroup of T . Moreover, any commutative compact connected Lie groupis of this form. o an orbit closure intersection (or equality) problem for a related action of the larger group, seeTheorem 8.2 for a precise statement.The results in this paper warrant the investigation of several interesting directions that we leavefor future work, some of which we will discuss in Section 9.1.4. Further motivation and algorithmic applications.
As we saw above, orbit problems arerelated to a great number of applications. Despite significant progress, for general reductive groupactions it is still an open problem to design fast algorithms for these problems. Our results fullyresolve the situation in the case of torus actions and also show how to overcome barriers thathad previously been pointed out in the literature [IMW17, GIM + w for the completebipartite graph on 2 n labeled vertices ( n on each side): the weight w ( e ) of an edge e is assumedto be a rational number, encoded in binary. We define the weight w ( M ) of a perfect matching M of G as the sum of the weights of the edges occurring in M . Problem 1.7.
Given edge weights w and w ′ as above, decide whether they assign the same weightto every perfect matching M of G .Perhaps surprisingly, this problem can be reformulated as an orbit intersection problem fora torus action (see below). Therefore, Theorem 1.2 implies that Problem 1.7 can be solved inpolynomial time. This insight seems far from being obvious!The relevant torus action here results from from matrix scaling, which has been widely studiedand has many applications; see [Sin64] and [CMTV17] for more recent developments. ConsiderST n := { ( t , . . . , t n ) ∈ C × | t · · · t n = 1 } , which is isomorphic to the algebraic torus ( C × ) n − . Welet ST n × ST n act on Mat n ( C ) by left-right multipliation as follows:(1.2) (( t , . . . , t n ) , ( s , . . . , s n )) · ( v ij ) := ( t i v ij s j ) ij . Moreover, we shall identify the edge weights w ij , where i, j ∈ [ n ], with the matrix v w = (2 w ij ) ∈ Mat n ( C ). Then one can show that the answer to Problem 1.7 is affirmative if and only if theorbit closures of v w and v w ′ in Mat n ( C ) intersect. This follows from Mumford’s theorem mentionedearlier, along with the fact that the invariant polynomials for this action are generated by theperfect matchings, namely the monomials f π = x ,π (1) · · · x n,π ( n ) where π ∈ S n ranges over thepermutations [LM99, Theorem 3]. Indeed, multiplying entries of v w is the same as summingthe corresponding edge weights in the exponent, hence f π ( v w ) = 2 w ( M ) , where M is the perfectmatching defined by the permutation π .We briefly comment on the 3-dimensional generalization of this action. Here, ST n × ST n × ST n acts on 3-tensors in C n ⊗ C n ⊗ C n in the natural way:(( t , . . . , t n ) , ( s , . . . , s n ) , ( u , . . . , u n )) · ( v ijk ) = ( t i s j u k v ijk ) ijk . In this case, any system of generating polynomial invariants must include the (maximum) 3-dimensional matching monomials f π,τ = x ,π (1) ,τ (1) · · · x ,π ( n ) ,τ ( n ) for π, τ ∈ S n , which led to thebarrier result for torus actions in [GIM + separating As explained in footnote 6, our results also hold for input of this form, where the w ij are specified in binary. olynomial invariants (whenever they exists) as well as to construct systems of generating invariant Laurent polynomial or rational invariants.Our second example concerns a connection to dynamical systems. Consider a (massless) cue ballon a billiard table (assumed to be square to simplify the discussion). We can ask: Problem 1.8.
If we hit the cue ball at a given angle, will its trajectory end up in a pocket?It is well-known, and easy to see, that one can map trajectories on an ordinary billiard withreflecting boundaries to a billiard of twice the size with periodic boundaries, say ( R / π Z ) . Thetrajectory of the ball depends fundamentally on the angle or slope. If the slope is irrational, thenthe trajectory will be dense, so the answer to Problem 1.8 is trivially yes. Otherwise, the trajectorywill be periodic and the problem is nontrivial. We can model it as an orbit problem as follows. Letthe compact torus S act on C by t · ( x, y ) := ( t p x, t q y ) , where s = qp is the slope by which we hit the ball. We can identify points ( θ, ν ) on the periodicbilliard with points ( e iθ , e iν ) ∈ C . In this way, Problem 1.8 reduces to a constant number of orbitequality problems for this action (one for each pocket). While the problem is certainly easy tosolve by a variety of methods, one can ask analogous questions for billiards in n > d -dimensional hyperplane worth of allowed cue directions. Such generalizationssimilarly correspond to orbit problems for compact tori (S ) d on some C n , and they can all besolved in polynomial time by using Corollary 1.6.1.5. Organization of the paper.
In Section 2, we give an introduction to basic results in invarianttheory that we will need to establish our results. In Section 3, we focus on tori, their representations,and their invariants. In particular, we will show that the faces of a natural convex polyhedral“Newton cone” are in one-to-one correspondence with the orbits in an orbit closure, which will bean important ingredient later on.In Section 4, we discuss the definition and computation of suitable rational invariants. Asmentioned above, our key result is that for fixed support, a small generating set of invariantLaurent monomials can be computed efficiently. This result, which is Theorem 1.3, is at the heartof our algorithms, and also of independent interest. We achieve this using Smith normal forms.As an easy consequence, this also implies that we can efficiently compute a small generating set ofrational invariants for a given representation, that is, Corollary 1.4.In Section 5, we explain how to use the results of the preceding section to solve the orbit equalityproblem in polynomial time. This establishes part (1) of Theorem 1.2. Here we rely on knownresults for testing if a given Laurent monomial (of possibly exponential degree) evaluates to thesame value on two given vectors, and we present a brief sketch for completeness.In Sections 6 and 7, we show how to solve the orbit closure intersection and containment problemsby reducing them to orbit equality. This establishes parts (2) and (3) of Theorem 1.2. Here weuse the polyhedral description of the structure of orbit closures as furnished by the Newton cone.Furthermore, we show that given two points whose orbit closures do not intersect, we can efficientlyconstruct a separating monomial invariant as a “witness”. This proves Corollary 1.5.In Section 8, we show how to solve the orbit equality problem for compact tori. This establishesCorollary 1.6. We also give, for general reductive groups G , a reduction from orbit equality for amaximally compact subgroup K ⊆ G to orbit equality and orbit closure intersection for G .In Section 9, we summarize our results and discuss some important open problems and futuredirections. Conventions.
In this paper, sometimes we work with monomials and sometimes with Laurentmonomials. Unless we use the prefix “Laurent”, by a monomial, we mean Q j x c j j where c j ∈ Z ≥ , .e., all exponents are non-negative. Whenever exponents are allowed to be negative, we will becareful to specify that it is a Laurent monomial.2. Preliminaries of invariant theory
We will briefly recall the main results in invariant theory that are relevant for us (see [Kra84,Dol03, DK15, MFK94] for details). We will take our ground field to be C , the field of complexnumbers, for simplicity. However, much of this theory works for any algebraically closed field. Fora (finite-dimensional) vector space V , we denote by C [ V ] the ring of polynomial functions on V .For our purposes, if V is the standard vector space C n , then C [ V ] = C [ x , . . . , x n ], the polynomialring in n variables, where x i is to be interpreted as the i th coordinate function.Let G be an algebraic group, i.e., it has the structure of an algebraic variety (not necessarilyirreducible) such that the multiplication map m : G × G → G and the inverse map ι : G → G aremorphisms of varieties. A morphism of algebraic groups ρ : G → GL( V ) is called a rationalrepresentation of G . We write gv or g · v for ρ ( g ) v . For a point v ∈ V , its orbit O v (or O G,v whenthe group is not clear from context) is the set of all points that can be reached from v by the actionof an element of the group, i.e., O v := { gv | g ∈ G } . We denote by O v the closure of the orbit O v . The closure is to be taken either with respect to theEuclidean topology or the Zariski topology. Indeed, the closures in both topologies coincide, a well-known fact that relies on a fundamental result in algebraic geometry due to Chevalley (see [Mum88,I. § f ∈ C [ V ] is called invariant if it is oblivious to the group action,i.e., f ( gv ) = f ( v ) for all g ∈ G , v ∈ V . The collection of all invariant polynomials forms a subring C [ V ] G := { f ∈ C [ V ] | ∀ g ∈ G, v ∈ V f ( gv ) = f ( v ) } . One key observation is that invariant functions are constant along orbits and hence constant alongorbit closures as well. Hence, if the orbit closures of two points intersect, then they cannot bedistinguished by an invariant function. The converse was proved by Mumford for a special classof groups called reductive groups [MFK94] (see also [DK15, Corollary 2.3.8]). An algebraic group G is called reductive if every rational representation is a direct sum of irreducible representations,wherein a representation is called irreducible if it has no non-trivial subrepresentations. Examples ofreductive groups include SL n , GL n , Sp n , O n , finite groups, and most importantly for us, tori (whichwe define formally in the next section), as well as direct products thereof. Reductive groupshave played a central role for a number of mathematical fields for over a century. A particularlyimportant result in the invariant theory of reductive groups is that invariant rings are finitelygenerated [Hil90, Hil93, Wey39].To state Mumford’s result in the generality we need, we will define rational actions on varieties(a notion that naturally generalizes rational representations). Let X be an algebraic variety andlet C [ X ] denote the ring of regular functions on X . A rational action of an algebraic group G on X is a morphism of varieties G × X → X, ( g, x ) g · x satisfying g · ( g ′ · x ) = ( gg ′ ) · x and e · x = x for all x ∈ X , g, g ′ ∈ G . As in the vector space case, we denote the orbit of a vector v ∈ X by O v . A morphism of varieties simply means that in local coordinates the map is given by ratios of polynomials. Forconcreteness, the reader may simply think of an algebraic group as a matrix group, i.e., a subgroup of GL n ( C ) thatis described as the zero locus of a collection of polynomials. One can interpret this action as the action of the subgroup ρ ( G ) ⊆ GL( V ) on V by matrix-vector multiplication,where ρ ( G ) is parametrized algebraically by an algebraic group G . The group B n of upper triangular n × n invertible matrices is a typical example of a group that is not reductive. heorem 2.1 (Mumford, [MFK94]) . Let G be a reductive group. Let X be an algebraic varietyand suppose we have a rational action of G on X . For v, w ∈ X we have O v ∩ O w = ∅ if and onlyif there exists f ∈ C [ X ] G such that f ( v ) = f ( w ) . Another well-known important structural result states that every orbit closure O v contains a unique closed orbit. Theorem 2.2.
Let ρ : G → GL( V ) be a rational representation of a reductive group G . Then:(1) For any v ∈ V , the orbit closure O v contains a unique closed orbit, that we denote by O e v .(2) If v, w ∈ V , then O v ∩ O w = ∅ ⇐⇒ O e v = O e w . Proof. (1) The first assertion is [DK15, Theorem 2.3.6].(2) For the second assertion, if the orbit closures O v and O w are disjoint, then so are the orbits O e v and O e w , which therefore must be different. Conversely, suppose O e v = O e w . Since these orbits areclosed, by Theorem 2.1, there is an invariant f ∈ C [ V ] G such that f ( e v ) = f ( e w ). By continuity, f ( v ) = f ( e v ) = f ( e w ) = f ( w ), which implies O v ∩ O w = ∅ by another application of Theorem 2.1. (cid:3) Part(2) of this theorem shows that the orbit closure intersection problem can be reduced to theorbit equality problem, provided we can compute the unique closed orbit O e v contained in O v . Wewill see in Section 6 that if the group G is a torus, this can be achieved in polynomial time.Another key result in understanding orbit closures is the Hilbert–Mumford criterion. A one-parameter subgroup of G is a morphism of algebraic groups σ : C × → G . For a representation of G on a vector space V , we say that a subset S ⊆ V is G -stable if g · s ∈ S for all g ∈ G , s ∈ S . Theorem 2.3 (Hilbert–Mumford criterion, [Hil93, MFK94]) . Let ρ : G → GL( V ) be a rationalrepresentation of a reductive group G . Suppose S ⊆ V is a G -stable closed subvariety of V andlet v ∈ V such that O v ∩ S = ∅ . Then there exists a one-parameter subgroup σ : C × → G such that lim ǫ → σ ( ǫ ) · v ∈ S . A particular use of the above theorem is to take S = { } or S = O ˜ v . When G is a torus, theset of one-parameter subgroups has the structure of a Z -lattice. We will discuss this further in thenext section.We end this section by introducing a key notion in invariant theory called the null cone , whosesignificance will become clear in later sections. For a collection F of polynomials in C [ V ], we denoteby V ( F ) their common zero locus in V . Definition 2.4 (Null cone) . Let ρ : G → GL( V ) be a rational representation of a reductive group G .Then the null cone is defined as N G ( V ) := N ( ρ ) := { v ∈ V | ∈ O v } . It can also be defined as the common zero locus of all invariant polynomials without constant part: N G ( V ) := N ( ρ ) := V ( [ d> C [ V ] Gd ) , where C [ V ] Gd denotes the space of invariant polynomials that are homogeneous of degree d . Theequivalence of the two definitions of the null cone follows from Theorem 2.1.3. Invariants and orbit closures of torus actions
Invariant theory for general reductive groups can get very complicated. However, for representa-tions of tori, that is, commutative connected reductive groups, a lot of the theory can be viewed as acombination of linear algebra and the study of convex polytopes. We will collect important resultsregarding torus actions in this section and refer the reader to [Weh93, DK15] for more details. All he results in this section are already known or can be deduced from the existing literature, andwe provide proof sketches for completeness. Note that tori are reductive groups, so the results ofthe previous section hold in this setting.We will first briefly recall torus actions and the notions of characters/weights, one-parametersubgroups and how weight matrices define a representation. Then, we give a linear algebraic de-scription of invariant rings by determining the monomials that are invariant. Then, we describea polyhedral perspective on orbits. In particular given a point v in the vector space of the repre-sentation, we define a polyhedral cone, called the Newton cone. The Newton cone can be used todetermine whether v is in the null cone and moreover we give a correspondence between the facesof the Newton cone to orbits in the orbit closure of v , which is crucial in understanding the orbitclosure containment problem.For this entire section, fix a torus T = ( C × ) d . Representations and invariants.
As described in Section 1.3, any representation of atorus T is a “scaling” action (after identifying V with C n by an appropriate choice of basis).Namely, each coordinate of v ∈ C n is multiplied by some (Laurent) monomial Q di =1 t λ i i for inte-gers λ i ∈ Z . These monomials (succinctly described by the so-called weight matrix, see below)together specify the representation. We now make this more precise.A 1-dimensional (rational) representation is called a character or a weight . Let X ( T ) denote theset of weights of T , which forms a group where the binary operation is (pointwise) multiplicationof functions. To each λ = ( λ , . . . , λ d ) ∈ Z d , we associate a weight, also denoted λ by slight abuseof notation, namely λ : T → C × , λ ( t ) = d Y i =1 t λ i i , which gives an identification of abelian groups Z d ∼ = X ( T ).Let ρ : T → GL( V ) be a (rational) representation of T where V is an n -dimensional vector space.We can choose a basis of V consisting of weight vectors, wherein a vector v ∈ V is called a weightvector of weight λ ∈ X ( T ) if t · v = λ ( t ) v for all t ∈ T . Once we have chosen a weight basis, usingthe identification X ( T ) ∼ = Z d , the corresponding n weights can be collected into a d × n matrixwith integer entries, which we call the weight matrix of the representation. Up to permutationof the columns, it is independent of the choice of weight basis, and it classifies the representationup to isomorphism. Concretely, a matrix M = ( m ij ) ∈ Mat d,n ( Z ) describes the representation ρ M : T → GL n ( C ) defined in (1.1). That is, for t = ( t , . . . , t d ) and v = ( v , . . . , v n ) ∈ C n , we have t · v = ρ M ( t ) v = d Y i =1 t m i i ! v , d Y i =1 t m i i ! v , . . . , d Y i =1 t m in i ! v n ! . The matrix M is the weight matrix for this action. The j th standard basis vector e j is a weightvector of weight m ( j ) = ( m j , m j , . . . , m dj ) ∈ Z d = X ( T ). Note that m ( j ) is the j th column vectorof M .For the rest of this section, we fix an n -dimensional representation ρ M : T → GL n ( C ) of thetorus T = ( C × ) d given by a weight matrix M ∈ Mat d,n ( Z ) with columns m ( j ) for j ∈ [ n ]. Thefollowing well-known result describes the invariant ring of this action (see, e.g., [DM20b, Section 3]): Proposition 3.1. (1) Let c ∈ Z n ≥ . A monomial x c = Q j x c j j is invariant if and only if P j c j m ( j ) = 0 ;(2) The invariant ring C [ x , . . . , x n ] T is spanned as a vector space by the invariant monomials. Any commutative connected reductive group is isomorphic to some ( C × ) d . Important examples include T d , thegroup of diagonal d × d invertible matrices and its subgroup ST d consisting of diagonal matrices with determinant 1. roof. For the action ρ of G on V , there is a natural induced action of G on the ring of polynomialfunctions C [ V ] defined by the formula g · f ( v ) := f ( ρ ( g ) − v ). Applying this for the action ρ M , weget an induced action of T on C [ x , . . . , x n ]. It is easy to compute this action: for a monomial x c and t ∈ T , we have t · x c = λ ( t ) − x c , where λ ∈ X ( T ) is the character corresponding to P j c j m ( j ) ∈ Z d .It follows that the monomials which are invariant are precisely the ones for which P j c j m ( j ) = 0,the trivial character, proving the first part. The second part follows from the observation that apolynomial is invariant if and only if each monomial that occurs in it is invariant. (cid:3) Part (1) of Proposition 3.1 shows that the invariant monomials are in one-to-one correspondencewith the nonnegative integer vectors in the kernel of the weight matrix. Accordingly, they form asemigroup. In general, such semigroups can have a large number of generators, which explains thedifficulty of using polynomial invariants [DHJ02]. Our key idea to obtain efficient algorithms willbe to instead consider invariant Laurent monomials, which form a lattice rather than a semigroup.We will return to this in Section 4.In turns out that the weights lead to a strong link to convex polyhedral geometry, which in turncharacterizes the orbits in an orbit closure. For this, we make the following definitions. The support of a vector v ∈ C n is defined as supp( v ) := { j ∈ [ n ] | v j = 0 } . Let us record some of the properties of the support. By dimension (of an orbit, orbit closure,algebraic group, etc), we mean the dimension of the underlying variety.
Lemma 3.2.
For v, w ∈ C n we have:(1) If O v = O w , then supp( v ) = supp( w ) .(2) If supp( v ) = supp( w ) , then dim O v = dim O w .(3) If w ∈ O v , then supp( w ) ⊆ supp( v ) . This inclusion is strict if and only if w ∈ O v \ O v .Proof. (1) is clear, since each coordinate simply gets rescaled by a nonzero number by the groupaction. For (2) we note that the stabilizer group stab( v ) of v only depends on supp( v ). The claimfollows using dim O v = d − dim stab( v ). For (3), the inclusion of supports holds since taking limitscan never increase the support. Finally, it is known [Hum75, § O v \ O v is a Zariski closedsubset of dimension strictly less than dim O v . Hence w ∈ O v \ O v implies dim O w < dim O v andtherefore supp( w ) ( supp( v ) by part (2). (cid:3) Newton cone and orbit closures.
We define the
Newton cone C ( v ) of a vector v ∈ C n to be the rational polyhedral cone generated by the weights corresponding to the indices in thesupport, that is, C ( v ) := n X j ∈ supp( v ) c j m ( j ) | c j ≥ o ⊆ R d . The lineality space of the cone C ( v ) is defined as L ( v ) := C ( v ) ∩ ( − C ( v )). Clearly, it is the largestlinear subspace contained in C ( v ). The cone C ( v ) is called pointed iff L ( v ) = 0. (Compare [Sch86]for the structure of polyhedral cones.)These notions are standard in geometric programming, which essentially studies optimizationproblems associated with torus actions, albeit often with a different representation and motiva-tion; see, e.g., [BLNW20] and references therein. The connection is particularly apparent anduseful in the study of polynomial capacities which have important applications to approximatecounting [LSW00, Gur04b].We will see that the Newton cone contains all the information about the orbits contained in anorbit closure. To start, we show that membership in the null cone can be characterized as follows. efine the essential support of a vector v ∈ V as(3.1) e-supp( v ) := { j ∈ supp( v ) | m ( j ) ∈ L ( v ) } . Lemma 3.3.
Let k ∈ supp( v ) . We have k ∈ e-supp( v ) if and only if there exists an invariantmonomial Q j ∈ supp( v ) x c j j with c j ∈ Z ≥ such that c k > .Proof. It is easy to see that m ( k ) ∈ L ( v ) if and only if there is a non-negative integral linearcombination P j ∈ supp( v ) c j m ( j ) = 0 with c k >
0. By Proposition 3.1, this is equivalent to theexistence of an invariant monomial Q j ∈ supp( v ) x c j j with c j ∈ Z ≥ such that c k > (cid:3) Corollary 3.4.
We have that v is in the null cone N ( ρ M ) if and only if e-supp( v ) = ∅ . Equivalently, v is in the null cone if and only if C ( v ) is pointed and m ( j ) = 0 for all j ∈ supp( v ).In fact, much more can be said. Let us first recall the notion of faces of polyhedral cones. If C ( v )is contained in a closed halfspace H + of R d bounded by a linear hyperplane H , then we call theintersection F = H ∩ C ( v ) a face of C ( v ) when it is non-empty. The cone itself is also considereda face of C ( v ): by definition, it is the largest face of C ( v ). On the other hand, each face of C ( v )must contain the lineality space L ( v ), which is therefore the smallest face of C ( v ).We will see shortly that the faces of C ( v ) are in bijective correspondence with the orbits containedin O v . For this, we need to introduce some more notation. For a subset J ⊆ supp( v ), we define the restriction v | J to be the vector with entries( v | J ) j = ( v j if j ∈ J, j -th coordinate. Let now F be a face of C ( v ) defined by a closed half-space H + = { y ∈ R d | ν · y ≥ } for some ν ∈ R d , that is, F = { y ∈ C ( V ) | ν · y = 0 } . Since C ( v ) is rational, we may assume that ν has integer components. We assign to F the subsetof indices S F := { j ∈ supp( v ) | m ( j ) ∈ F } and define v F := v | S F . Let us check that the orbit O v F of v F is contained in O v . The one-parametersubgroup σ : C × → T given by σ ( ǫ ) = ( ǫ ν , . . . , ǫ ν d ) satisfies(3.2) σ ( ǫ ) · v = ρ M ( σ ( ǫ )) v = ( ǫ ν · m (1) v , . . . , ǫ ν · m ( n ) v n ) . It follows that lim ǫ → σ ( ǫ ) · v = v F and hence v F ∈ O v . The same reasoning shows that v F ∈ O v F ′ if F is a face contained in the face F ′ .The following result is well known, see e.g., [Pop09, Example 1.3], but we sketch a proof forcompleteness. Proposition 3.5.
The map F O v F is a bijection between the set of faces of C ( v ) and the set oforbits contained in O v . Moreover, we have F ⊆ F ′ ⇐⇒ O v F ⊆ O v F ′ . The proof of surjectivity relies on a strengthening of the Hilbert–Mumford criterion (Theo-rem 2.3). Recall this states that if we consider a closed subset S that is stable under the groupaction and intersects the orbit closure of some point v , then there is a one-parameter subgroupthat will drive v to a point in S in the limit. However, a subtle point is that this requires S to beclosed. In general, orbits are not closed, so a point w could be in the orbit closure of a point v , butthe orbit of w may not be closed. In this case, Theorem 2.3 does not apply to S = O w , and indeedthe orbit of w need not be reachable from v by a limit of a one-parameter subgroup. The following heorem shows that for torus actions such a phenomenon does not happen. This crucial fact willalso prove useful for us algorithmically in Section 7. Theorem 3.6 ([Kra84], Kapitel III.2.2) . Let ρ : T → GL( V ) be a rational representation. Suppose v, w ∈ V are such that w ∈ O v . Then there exists a one-parameter subgroup σ : C × → T such that lim ǫ → σ ( ǫ ) · v ∈ O w . Before we prove Proposition 3.5, we discuss a bit about the structure of one-parameter subgroups.For each ν ∈ Z d , we define a one-parameter subgroup of T , namely σ : C × → T defined by σ ( ǫ ) =( ǫ ν , . . . , ǫ ν d ). Any one-parameter subgroup of T is of this form. This gives an identification ofabelian groups Z d ∼ = Y ( T ), where Y ( T ) denotes the collection of all one-parameter subgroups of T .We leave the proof of the following well known lemma to the reader. Lemma 3.7.
Let σ : C × → T be a one-parameter subgroup, so σ ( ǫ ) = ( ǫ ν , . . . , ǫ ν d ) for some ν ∈ Z d ,and let v ∈ C n .(1) The limit lim t → σ ( t ) · v exists if and only if m ( j ) · σ ≥ for all j ∈ supp( v ) .(2) If the limit exists, then lim t → σ ( t ) · v = v | S , where S = { j ∈ supp( v ) | m ( j ) · σ = 0 } .Proof of Proposition 3.5. We have already verified that O v F is an orbit contained in O v , hence F O v F is well-defined as a map from the set of faces of C ( v ) to the set of orbits contained in O v .To see that it is injective, note that F is the cone generated by supp( v F ) = S F . For surjectivity,let O w be an orbit contained in O v and σ : C × → T be a one-parameter subgroup as in Theorem 3.6.There is ν ∈ Z d such that σ ( ǫ ) = ( ǫ ν , . . . , ǫ ν d ). By Lemma 3.7, the existence of lim ǫ → σ ( ǫ ) · v means that ν · m ( j ) ≥ j ∈ supp( v ). In other words, C ( v ) is contained in the halfspace { y ∈ R d | ν · y ≥ } . Moreover, the limit equals v F , where F is the face F := { y ∈ C ( v ) | ν · y = 0 } of C ( v ). Therefore, v F ∈ O w , hence O v F = O w , and we have shown surjectivity.In order to show the remaining equivalence, recall that we argued below (3.2) that if F ⊆ F ′ then v F ∈ O v F ′ . The preceding argument also implies the converse. (cid:3) As an immediate consequence of Proposition 3.5, we get the following result, which not onlyreproves Lemma 3.3 but also characterizes the closed orbit in an orbit closure. For this, define e v := v | L ( v ) = v | e-supp( v ) . Corollary 3.8.
The orbit O ˜ v corresponding to the lineality space L ( v ) is contained in every orbitclosure contained in O v . Therefore, it is the unique closed orbit contained in O v .In particular, the orbit O v is closed if and only if C ( v ) = L ( v ) , i.e., C ( v ) equals its linear span.Moreover, v is in the null cone if and only if e-supp( v ) = ∅ . Generating Laurent polynomials and rational invariants
In this section, we discuss the computation of suitable rational invariants, which is the heart ofour algorithms, and the main novelty of this paper. As explained in the introduction, the startingpoint is the simple observation that two orbits can only be equal when they have the same support(Lemma 3.2). But once we restrict to vectors of fixed support, it is natural to consider a largerclass of invariants, namely Laurent polynomials, which are polynomials that can also have negativeexponents. In Section 4.1 we will see that the invariant Laurent polynomials for a given supportnaturally form a lattice that can be computed from the weight matrix. This allows us to give anefficient algorithm for computing small sets of generators. As a consequence, we can also efficientlycompute a system of generating rational invariants.For the rest of this section, we fix an n -dimensional representation ρ M : T → GL n ( C ) of thetorus T = ( C × ) d given by a weight matrix M ∈ Mat d,n ( Z ) with columns m ( j ) for j ∈ [ n ]. .1. Invariant Laurent polynomials.
For S ⊆ [ n ], consider the set of vectors with support S ,that is, the variety(4.1) X S = { v ∈ C n | supp( v ) = S } = { v ∈ C n | v j = 0 if and only if j ∈ S } . The ring of regular functions on X S , denoted C [ X S ], is naturally identified with the ring of Laurentpolynomials in variables { x j } j ∈ S . That is, C [ X S ] = C [ x j , x − j | j ∈ S ] . We observe that ρ M restricts to an action of T on X S , and induces an action on C [ X S ]. Theproposition below shows that the algebra C [ X S ] T of invariant Laurent polynomials can be succinctlydescribed in terms of the lattice(4.2) L S = n c ∈ Z S | X j ∈ S c j m ( j ) = 0 o = ker( M S ) ∩ Z | S | , where Z S := { c ∈ R n | c j = 0 for all j S } ∼ = Z | S | , and M S denotes the submatrix of the weightmatrix M , obtained by removing all columns except those labeled by S . Proposition 4.1. (1) Let c ∈ Z S . A Laurent monomial x c = Q j ∈ S x c j j is invariant if and onlyif c ∈ L S .(2) The algebra of invariant Laurent polynomials C [ X S ] T is spanned as a vector space by theinvariant Laurent monomials.(3) If { c (1) , c (2) , . . . , c ( r ) } is a lattice basis of L S , then C [ X S ] T is generated as an algebra by theinvariant Laurent monomials { x c (1) , . . . , x c ( r ) } .Proof. The first two parts are shown using an argument similar to the proof of Proposition 3.1.The third statement is an immediate consequence. (cid:3)
It is instructive to compare this with the discussion below Proposition 3.1, where we saw thatthe invariant polynomials are similarly described by the semigroup of nonnegative vectors in thekernel of the weight matrix. By working with vectors of fixed support, we instead obtain a naturallattice structure, which simplifies the situation considerably. For example, the lattice L S and hencethe algebra of invariant Laurent polynomials C [ X S ] T have at most | S | ≤ n generators – in starkcontrast to the situation for invariant polynomials.We now discuss how to compute lattice bases as in Proposition 4.1. It is well known that everyinteger matrix M can be diagonalized by multiplying from left and right with unimodular matrices.This is known as the Smith normal form [Smi61]. The Smith normal form can be computed inpolynomial time [KB79]. We record these facts in the following theorem.
Theorem 4.2 (Smith normal form) . Let M ∈ Mat d,n ( Z ) . Then, there exist unimodular matrices U ∈ Mat d,d ( Z ) , W ∈ Mat n,n ( Z ) such that U M W = α . . . α . . .
00 0 . . . α r ...... ... . . . . . . and the diagonal elements satisfy α i | α i +1 for i = 1 , , . . . , r − , where r equals the rank of M .The matrix U M W is unique and called the Smith normal form of M . oreover, if the bit-lengths of the entries of M are bounded by b , then the matrices U , W ,and U M W can be computed in poly( d, n, b ) -time. Using the Smith normal form it is easy to compute a basis of the lattice L S . We state this inthe following algorithm and corollary. Algorithm 4.3.
Computation of a basis of the lattice of invariant Laurent monomials:
Input: M ∈ Mat d,n ( Z ) and S ⊆ [ n ]. Step 1:
Compute the submatrix M S of M obtained by deleting all columns except those in S . Step 2:
Compute the Smith normal form
U M S W of M S (as in Theorem 4.2). Step 3:
Return { w ( r +1) , w ( r +2) , . . . , w ( n ) } , where w ( j ) denotes the j th column of W . Corollary 4.4.
Let M ∈ Mat d,n ( Z ) and S ⊆ [ n ] , and suppose the bit-lengths of the entries of M are bounded by b . Then Algorithm 4.3 computes a basis for the lattice L S defined in (4.2) in poly( d, n, b ) -time. In particular, each w ( j ) has bit-length poly( d, n, b ) . Alternatively, one can use lattice algorithms; we refer the interested reader to [GLS93, Corol-lary 5.4.10].
Remark 4.5.
It is easy to see that given an exponent vector c = ( c , . . . , c n ) ∈ Z n ≥ , where thebit-lengths of the c i s are bounded by b , an arithmetic circuit computing the monomial x c of sizepoly( n, b ) can be constructed in poly( n, b )-time. Similarly, if c ∈ Z n , an arithmetic circuit withdivision computing the Laurent monomial x c can be constructed in poly( n, b )-time. Proof of Theorem 1.3.
This follows from Proposition 4.1, Corollary 4.4, and Remark 4.5. (cid:3)
Rational invariants.
In the remainder of this section we will discuss rational invariants.For V = C n , recall that C [ V ] = C [ x , . . . , x n ] is the polynomial ring in n variables. Let C ( V ) = C ( x , . . . , x n ) the field of rational functions (its fraction field). In other words, any element in C ( V )is a ratio of two polynomials. The action of T on C [ V ] extends to C ( V ). Then C ( V ) T is the fieldof rational invariants . Clearly, any invariant Laurent polynomial is a rational invariant, but theconverse need not be the case.Nevertheless, we can show that the invariant Laurent polynomials in all variables (that is, forsupport S = [ n ]) generate the rational invariants as a field. Proposition 4.6.
Let A := C [ X [ n ] ] = C [ x , x − , . . . , x n , x − n ] T denote the algebra of invariantLaurent polynomials, and let F := C ( x , . . . , x n ) T denote the field of rational invariants. Then, A generates F as a field, i.e., the field of fractions of A is F .Proof. Let f ∈ F × and write f = pq , where p, q ∈ C [ x , . . . , x n ] have no common factors. Since f isinvariant, we have for any t ∈ T that t · pt · q = t · f = f = pq . Accordingly, t · p = α ( t ) p and t · q = α ( t ) q for some α ( t ) ∈ C × . Thus, p and q span one-dimensionalrepresentations. This in turn implies that α : T → C × is a character, as discussed in Section 3.1,and further that p (and also q ) is a sum of monomials with the same weight, i.e., p = P e p e x e suchthat t · x e = α ( t ) x e for p e = 0. In particular, f e = qx e is a Laurent polynomial invariant if p e = 0,and we can write f = pq = X e p e x e q = X e p e f e , which concludes the proof. (cid:3) s a direct consequence, any system of generating invariant Laurent polynomials (as an algebra)also serves as a system of generating rational invariants (as a field extension of C ). Thus we obtain: Proof of Corollary 1.4.
This follows from Theorem 1.3 (with S = [ n ]) and Proposition 4.6. (cid:3) Orbit equality problem
In this section, we will give a polynomial time algorithm for the orbit equality problem. Giventwo points, the strategy is to compute a small collection of invariant Laurent monomials (usingthe result of Section 4) whose evaluations at the two given points will determine whether the twopoints are in the same orbit. The efficient testing of whether two Laurent monomials evaluate tothe same value actually requires an idea: this has already been studied in the literature and webriefly sketch in Section 5.1 how to do this.We still assume an n -dimensional representation ρ M : T → GL n ( C ) of the torus T = ( C × ) d givenby a weight matrix M ∈ Mat d,n ( Z ) with columns m ( j ) for j ∈ [ n ].In general, invariants can only decide orbit closure intersection, not orbit equality. However, thecrucial point is that in the varieties (4.1) consisting of vectors of fixed support any T -orbit is closed. Proposition 5.1.
Let S ∈ [ n ] , X S be the variety defined in (4.1) , and v ∈ X S . Then the orbit O v is a closed subset of X S .Proof. By Lemma 3.2 (3) we have O v = O v ∩ X S which implies that the orbits are closed in X S . (cid:3) Orbit equality in V can always be reduced to orbit equality in some X S , since equality of supportsis a necessary condition (Lemma 3.2 (1)). The importance of the above result is that the latterorbit equality and orbit closure intersection are equivalent in X S . Together with Theorem 2.1 weobtain the following result. Corollary 5.2.
Suppose supp( v ) = supp( w ) = S . Then, O v = O w if and only if there is aninvariant Laurent monomial f = Q j ∈ S x c j j such that f ( v ) = f ( w ) . Thus, we obtain the following algorithm for the orbit equality problem.
Algorithm 5.3.
Deciding orbit equality:
Input: M ∈ Mat d,n ( Z ) and v, w ∈ Q ( i ) n . Step 1:
Check if supp( v ) = supp( w ). If not, O v = O w , so we can stop. Step 2:
Use Algorithm 4.3 to compute a lattice basis B for the lattice L S defined in (4.2). Step 3:
For each e ∈ B , we check if v e = w e (as described in Section 5.1 below).If they are all equal, then O v = O w . Else, O v = O w . Proof of Theorem 1.2, part (1).
The correctness of Algorithm 5.3 follows from Proposition 4.1 andCorollary 5.2. We now analyze its runtime. Clearly, the first step can be implemented efficiently.For the second step, we can appeal to Corollary 4.4. For step 3, we first observe that, again byCorollary 4.4, the exponents e have bit-length poly( d, n, b ). Then Proposition 5.5 below shows thatthis step can also be implemented in time poly( d, n, b ). (cid:3) Laurent monomial equivalence.
We now discuss how to test if a Laurent monomial x e evaluates to the same value at two points v and w . In our context, where each component e j of the exponent vector e = ( e , . . . , e n ) has poly-sized bit-lengths, it is unreasonable to evaluatethe Laurent monomials explicitly, because the answer may very well require exponentially largebit-length. Yet, it is possible to check if v e = w e efficiently. We describe a simple algorithm basedon g.c.d.’s, which has appeared before (see, for example, [ESY14]) in the case where the entriesof v and w are in Z (or equivalently Q ). The result is much older; for example, it follows from he results in [BDS90], as mentioned in [Ge93], which gives a generalization to number fields. Here we present a short self-contained proof and then follow up with the rather simple extensionto Gaussian rationals.
Lemma 5.4.
Suppose a , . . . , a k , b , . . . , b r ∈ Q and e , . . . , e k , f , . . . , f r ∈ Z have bit-lengths atmost s . Then, in poly( k, r, s ) -time, we can decide if Q ki =1 a e i i = Q rj =1 b f j j .Proof. By clearing denominators, we may assume that a , . . . , a k , b , . . . , b r are integers. By movingterms to the other side, we can further assume w.l.o.g.that all e i , f j ≥
0. Pick some a l and some b m that are not coprime. Then, consider d = gcd( a l , b m ) ≥
2. W.l.o.g., we can assume e l ≥ f m . Then,test if d e l − f m ( a ′ l ) e l Q i = l a e i i = ( b ′ m ) f m Q j = m b f j j , where a ′ l = a l /d and b ′ m = b m /d . This is an iterativeprocedure which stops when each a i is coprime to b j . At which point, unless all a i ’s and b j ’s areequal to 1, both sides cannot be equal.The question is how long does such an iterative procedure take. Consider the quantity P := | a · · · a k b · · · b r | . After applying one step, the resulting quantity P ′ satisfies P ′ = P/d ≤ P/ P is 2 poly( k,r,s ) -sized, there are at most a polynomial number of iterative steps.Hence, the entire procedure takes poly( k, r, s )-time. (cid:3) An analogous result with the same proof holds for the ring Z [ i ] of Gaussian integers and its quo-tient field Q ( i ) of Gaussian rationals, using that this ring has unique factorization into irreducibleelements. In the following proposition, we assume that a Gaussian rational a = α + iβ ∈ Q ( i ) isdescribed by giving the encodings of α and β in binary. Proposition 5.5.
Suppose a , . . . , a k , b , . . . , b r ∈ Q ( i ) and e , . . . , e k , f , . . . , f r ∈ Z all have bit-lengths bounded by s . Then, in poly( k, r, s ) -time, we can decide if Q ki =1 a e i i = Q rj =1 b f j j . Remark 5.6.
For computational purposes, in many instances, numbers are described by their‘floating point’ representations. The floating point description of a Gaussian rational a ∈ Q ( i ) isdescribed by giving the binary encodings of α, β ∈ Q and p ∈ Z such that a = ( α + iβ )2 p . If weassume that a , . . . , a k , b , . . . , b r ∈ Q ( i ) in the proposition above are given by their floating pointdescriptions, we can still decide monomial equivalence in polynomial time. Indeed, if we write each a j = ( α j + iβ j )2 p j and b j = ( γ j + iδ j )2 q j , then deciding whether Q kj =1 a e j j = Q rj =1 b f j j simplifies todeciding if k Y j =1 ( α j + iβ j ) e j · P kj =1 e j p j = r Y j =1 ( γ j + iδ j ) f j · P rj =1 f j q j , which can again be interpreted as an instance of Proposition 5.5 and hence can be checked inpolynomial time. Since all other computations in our algorithms only involve supports of vectors,it follows that all results in this paper generalize to this input model, as claimed in footnote 6.An even easier special case arises for numbers of the form a = 2 p , with p ∈ Q specified by itsbinary encoding, as in the perfect matching application discussed in Section 1.4. Indeed, if a j = 2 p j and b j = 2 q j for j ∈ [ n ], then deciding whether Q kj =1 a e j j = Q rj =1 b f j j simply amounts to verifyingwhether P nj =1 p j e j = P nj =1 q j f j , which is clearly possible in polynomial time.6. Orbit closure intersection and explicit separating invariants
In this section, we discuss how to solve the orbit closure intersection problem in polynomial timeby efficiently reducing it to the orbit equality problem. The problem of orbit closure intersectionhas a manifestly analytic point of view, but also an algebraic point of view by Mumford’s theorem, In particular Ge’s result [Ge93] implies that Theorem 1.2 extends to the case where the entries of v and w aretaken from some algebraic number field. heorem 2.1. In other words, when orbit closures of two points do not intersect, there is aninvariant polynomial that takes different values on both points, serving as a “witness” to the factthat the orbit closures do not intersect. Accordingly, given two vectors whose orbit closures donot intersect, we also explain how to efficiently construct an arithmetic circuit which computes aninvariant monomial separating the two vectors.6.1. Reduction to orbit equality.
The key idea is the following. Recall from Theorem 2.2 thatany orbit closure O v contains as unique closed orbit O ˜ v , and that two orbit closures intersect ifand only if they contain the same closed orbit. In Corollary 3.8, we showed that the unique closedorbit has a concrete polyhedral characterization: we can take ˜ v = v | e-supp( v ) , the restriction of thevector v to its essential support. Accordingly, the map v e v provides a reduction of the orbitclosure intersection problem for ρ M to the the orbit equality problem for ρ M . The following lemmashows that the essential support (and hence the reduction map) can be computed in polynomialtime by using linear programming. Lemma 6.1.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d , andlet v ∈ C n . For k ∈ supp( v ) , we have k ∈ e-supp( v ) if and only if there is a non-negative linearcombination P j ∈ supp( v ) c j m ( j ) = 0 such that c k > . If the bit-lengths of the entries of M arebounded by b , the latter can be decided in poly( d, n, b ) -time by using linear programming.Proof. The characterization follows from Proposition 3.1 and Lemma 3.3. It amounts to a basicdecisional problem of linear programming, which is well known to be solvable in polynomial time,see [GLS93]. (cid:3)
The above proof also shows that a nonvanishing invariant monomial as in Lemma 6.1 can becomputed in polynomial time. As explained above, we arrive at the following algorithm and results.
Algorithm 6.2.
Reduction of orbit closure intersection to orbit equality:
Input: M ∈ Mat d,n ( Z ) , v, w ∈ Q ( i ) n . Step 1:
Compute e-supp( v ) in the following way: For each k ∈ supp( v ), use linear programmingto determine if there is a non-negative linear combination P j ∈ supp( v ) c j m ( j ) = 0 with c k > v ) consists of all k ∈ supp( v ) for which this is the case. Step 2:
Compute e-supp( w ) in the same way. Step 3:
Return ˜ v = v | e-supp( v ) and ˜ w = w | e-supp( w ) . Corollary 6.3.
Let M ∈ Mat d,n ( Z ) describe an n -dimensional representation of T = ( C × ) d .Further, let v, w ∈ Q ( i ) n and assume the bit-lengths of the entries of M, v , and w are bounded by b .Then there is a poly( d, n, b ) -time reduction that reduces the problem of deciding O v ∩ O w = ∅ to theproblem of deciding if O e v = O e w , where e v and e w have bit-lengths bounded by b .Proof of Theorem 1.2, part (2) . This follows from part (1), combined with Corollary 6.3. (cid:3)
Explicit separating invariant.
For torus actions, our reduction of orbit closure intersectionto orbit equality will give us an invariant Laurent monomial that takes different values on the twopoints. But a separating invariant Laurent monomial itself does not serve as a witness (at least notnaively, one needs further properties about the support of the Laurent monomial for it to serve asa witness). We now prove Corollary 1.5, which asserts that given two vectors we can neverthelessefficiently construct an arithmetic circuit which computes an invariant monomial separating them.
Proof of Corollary 1.5.
We already noted that, by linear programming, we can compute the essen-tial supports of v and w in poly( d, n, b )-time. We distinguish two cases. Case 1: e-supp( v ) = e-supp( w ) uppose k ∈ e-supp( v ) \ e-supp( w ) without loss of generality. By Lemma 3.3 there is an invariantmonomial f = Q j ∈ supp( v ) x c j j such that c k >
0. Let us verify that f ( v ) = f ( w ). We clearlyhave f ( v ) = 0. On the other hand, f ( w ) = f ( e w ) = 0, since e w ∈ O w , but k is not contained insupp( e w ) = e-supp( w ). So we indeed have f ( v ) = f ( w ). In addition, we can find ( c , . . . , c n ) inpoly( d, n, b )-time by linear programming (Lemma 6.1), so we can construct an arithmetic circuitfor f in poly( d, n, b )-time by Remark 4.5. Case 2: e-supp( v ) = e-supp( w )Let S := e-supp( v ) = e-supp( w ). We assume that O v ∩ O w = ∅ , which implies O e v ∩ O e w = ∅ . Thus, byCorollary 5.2, there is an invariant Laurent monomial f = x e with the property that f ( e v ) = f ( e w ),and hence f ( v ) = f ( w ). Just like in Algorithm 5.3, we can in poly( d, n, b )-time compute such anexponent vector e ∈ Z n , with bit-length of the e i bounded above by poly( d, n, b ).Our goal is to produce an invariant monomial that separates v and w , so we need to modify f so as to get rid of the negative exponents. In the process, we must ensure that the bit-length ofthe circuit does not explode. By Lemma 3.3, for each k ∈ S , there exists c ( k ) ∈ Z n ≥ such that P j ∈ supp( v ) c ( k ) j m ( j ) = 0 and c ( k ) k >
0. We can compute c ( k ) in poly( d, n, b )-time by linear program-ming. Let m k = x c ( k ) denote the corresponding invariant monomial. Put S − := { j ∈ S | e j < } .If m j ( v ) = m j ( w ) for some j ∈ S − , then m j is an explicit separating invariant monomial and we aredone by Remark 4.5. Assume now m j ( v ) = m j ( w ) for all j ∈ S − . Then e f := x d := f · Q j ∈ S − m − e j j is a Laurent monomial that separates v and w . We verify now that the exponent vector d hasnon-negative entries. By construction, we have for k ∈ S − , d k = e k + ( − e k ) c ( k ) k + X j ∈ S − ,j = k ( − e j ) · c ( j ) k ≥ , since e k < e j < j ∈ S − , while c ( k ) k ≥
1, and c ( j ) k ≥
0. For k ∈ [ n ] \ S − , we have d k = e k + X j ∈ S − ( − e j ) · c ( j ) k ≥ , since e k ≥ k ∈ S \ S − and e k = 0 for k S , while e j < j ∈ S − . Altogether, we haveshown that indeed all components of d are non-negative. We finally note that d can be computed inpolynomial time, in particular, it has bit-length poly( d, n, b ). So by Remark 4.5, we can constructan arithmetic circuit of size poly( d, n, b ) that computes e f in poly( d, n, b )-time. (cid:3) Orbit closure containment
In this section, we discuss how to solve the the orbit closure containment problem in polynomialtime by efficiently reducing it to the orbit equality problem.The notion of orbit closure containment is in general quite tricky to capture. Polynomial invari-ants do not suffice, since two orbit closures can intersect (hence all polynomial invariants agree)with neither being contained in the other – this is precisely the difference between the orbit closureintersection and the orbit closure containment problem. Instead, the key idea for the reductioncomes from one-parameter subgroups. We already discussed in Section 3 that if w ∈ O v then O w can be reached from v by a one-parameter subgroup. The following proposition gives a concretepolyhedral description of the relevant one-parameter subgroups. Lemma 7.1.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d , andlet v, w ∈ C n . Then w ∈ O v if and only if there exists a one-parameter subgroup σ : C × → T , so σ ( ǫ ) = ( ǫ ν , . . . , ǫ ν d ) for some ν ∈ Z d , such that(1) { j ∈ supp( v ) | m ( j ) · ν = 0 } = supp( w ) and m ( k ) · ν > for all k ∈ supp( v ) \ supp( w ) ;(2) O ( v | supp( w ) ) = O w . roof. If w ∈ O v , then by Theorem 3.6, we know that there is a one-parameter subgroup σ suchthat lim t → σ ( t ) v ∈ O w . In particular this implies that lim t → σ ( t ) v has the same support as w andhas the same orbit as w . Now, both (1) and (2) follow from Lemma 3.7.For the converse, note that, again by Lemma 3.7, (1) implies that lim t → σ ( t ) v = v | supp( w ) ∈ O v ,hence it follows that O w = O ( v | supp( w ) ) ⊆ O v by (2). (cid:3) Now, we can give our algorithm to test if w is in the orbit closure of v . Algorithm 7.2.
Orbit closure containment:
Input: M ∈ Mat d,n ( Z ) and v, w ∈ Q ( i ) n . Step 1:
Check if supp( w ) ⊆ supp( v ). If not, w / ∈ O v , so we can stop. Step 2:
Using linear programming, determine whether there exists a solution y ∈ R d to thecollection of linear equalities m ( j ) · ν = 0 for each j ∈ supp( w ) and linear inequalities m ( k ) · ν > k ∈ supp( v ) \ supp( w ). If there is no solution, then w / ∈ O v , so we canstop. Step 3:
Use Algorithm 5.3 check whether O ( v | supp( w ) ) = O w . If yes, then w ∈ O v . Else, it is not. Proof of Theorem 1.2, part (3).
The correctness of Algorithm 7.2 follows from Lemma 7.1. Indeed,condition (1) in the lemma is satisfied if and only if the algorithm passes the first two steps, andthen condition (2) is tested in the last step.We still need to argue about the efficiency of the algorithm. Clearly, step 1 can be done in lineartime. Step 2 can be done in poly( d, n, b )-time by linear programming. Step 3 appeals to the orbitequality problem, which by part (1) of the theorem can be done in poly( d, n, b )-time. (cid:3) Orbit problems for compact tori
So far, we have studied orbit problems for algebraic tori, that is, groups of the form T = ( C × ) d .In this section we consider the groups K = (S ) d , where S = { z ∈ C × | | z | = 1 } . Such groups areoften called compact tori . Indeed, any commutative compact connected Lie group is of this form.Besides the fundamental algorithmic interest in this setting, it is also important in applications.For example, in physics, symmetries are often given by compact group actions, such as compacttori [GS90, Aud12]. We give further complexity-theoretic motivation below.The compactness implies that orbits are closed and so the three problems in Problem 1.1 coincide.In this section, we show how to solve the orbit equality problem for a compact torus by reducingit to orbit equality for the corresponding algebraic torus. Subsequently, we give an alternativereduction that works not only for tori but in fact for any connected reductive group such as SL n .To start, we note that it is known that any (continuous) finite-dimensional representation of K = (S ) d extends to a representation of T = ( C × ) d [Wey39]. In particular, representations canbe specified as before by a weight matrix M ∈ Mat d,n ( Z ). Then we have the following result: Proposition 8.1.
Let M ∈ Mat d,n ( Z ) define an n -dimensional representation of T = ( C × ) d and K = (S ) d . Let v, w ∈ C n . Then, O K,v = O K,w if and only if O T,v = O T,w and | v j | = | w j | for all j .Proof. Since K ⊆ T , it is clear that if O K,v = O K,w , then O T,v = O T,w and | v j | = | w j | for all j .Conversely, suppose O T,v = O T,w and | v j | = | w j | for all j . Then, there is some t ∈ T suchthat t · v = w . Write t = ( t , . . . , t d ) and write each t i = r i · e i θ i , with r i > θ i ∈ R . Then, it iseasy to see that we must have ( e i θ , . . . , e i θ d ) · v = w . Thus v and w are in the same K -orbit. (cid:3) Proof of Corollary 1.6.
We are given M ∈ Mat d,n ( Z ) and v, w ∈ Q ( i ) n . By the above proposition,we need to check if O T,v = O T,w and if | v j | = | w j | for all j . The former can be done in polynomialtime by Theorem 1.2 and the latter can clearly be done in polynomial time. (cid:3) efore proceeding we give some further context and motivation. Algorithms for the null conemembership problem (given a rational representation ρ : G → GL( V ) of a reductive group G and v ∈ V , decide if 0 ∈ O v ) based on optimization methods have emerged in recent years. They takeadvantage of the fact that 0 ∈ O v if and only if one can drive the norm to 0 along the orbit O v .This can be viewed as an optimization problem where one tries to minimize (infimize) the normalong the orbit. While this is not a convex optimization problem, it is geodesically convex by theKempf-Ness theory [KN79], which allows for many of the ideas to be modified appropriately. Asfar as the orbit closure intersection problem is concerned, the natural extension of this idea is asfollows: Given v, w ∈ V , first use an optimization algorithm to approximately find a point in eachorbit closure with minimal norm; let us call these points ˇ v , ˇ w . Then, appealing to the Kempf-Nesstheory again, we have that O v ∩ O w = ∅ if and only if ˇ v and ˇ w are in the same orbit for a maximalcompact subgroup K of G . In this way, the orbit closure intersection problem for G can be reducedto the orbit equality problem for the maximal compact subgroup K . In fact, for the so-calledleft-right action of SL n × SL n on matrix-tuples, this idea was carried out successfully to obtain apolynomial-time algorithm for orbit closure intersection [AZGL + G , the orbit equalityproblem for the maximal compact subgroup K ⊆ G is equivalent to an orbit intersection (orequality) problem for a related action of G ! As this result is not crucial to the rest of the paper andrequires significantly different background, we will be brief in our explanations. We denote by V ∗ the contragredient or dual representation of V . Theorem 8.2.
Let ρ : G → GL( V ) be a finite-dimensional representation of a connected reductivegroup G . Let K be a maximal compact subgroup of G , and h· , ·i be a K -invariant Hermitian innerproduct on V . For v ∈ V , let b v ∈ V ∗ be defined by b v ( w ) := h v, w i . Then, for v, w ∈ V , the followingare equivalent:(1) O K,v = O K,w ;(2) O G, ( v, b v ) = O G, ( w, b w ) in V ⊕ V ∗ ;(3) The G -orbit closures of ( v, b v ) and ( w, b w ) in V ⊕ V ∗ intersect.Proof. Let Lie( G ) ⊆ L ( V ) denote the Lie algebra of G . For any linear action of G on a vectorspace U , we get an induced action of Lie( G ) on U . Given a K -invariant Hermitian form h· , ·i on U , we define the so-called moment map µ U : U → Lie( G ) ∗ by the formula µ U ( u )( X ) = h u, X · u i for u ∈ U and X ∈ Lie( G ) (up to a scalar which is not relevant for our purposes). The celebratedKempf-Ness theorem says that if µ U ( u ) = 0 then the G -orbit of u is closed. Moreover, it assertsthat if u ′ ∈ U is another point such that µ U ( u ′ ) = 0, then O G,u = O G,u ′ if and only if O K,u = O K,u ′ .Applying the preceding to ( v, b v ) and ( w, b w ) in U = V ⊕ V ∗ , a simple calculation shows that themoment map vanishes at either point, so the two orbits are closed. This shows the equivalencebetween (2) and (3). The equivalence between (1) and (2) follows immediately from the secondpart of the Kempf-Ness theorem, using that k b v = c kv for any k ∈ K , since K acts unitarily. (cid:3) Concluding remarks, future directions, and open problems
To better understand the context of our results and their potential impact on future progress, webriefly discuss some results in literature and then suggest further research directions. In very highlevel, we feel that the following aspects are highlighted by this work: the relative power and interplaybetween algebraic and analytical algorithms, the importance of understanding commutative actionsas a stepping stone towards understanding general actions, the role of rational (as opposed topolynomial) invariants, and the subtlety of “no go” results, which evidently can be surpassed. here has been an explosion of interest over the last decade in understanding invariant theoryfrom a complexity theoretic perspective (we survey some of this literature in the introduction).This rapidly developing field can be seen as an endeavor to classifying computational problems ininvariant theory according to their difficulty, finding efficient algorithms whenever possible, as wellas connecting to applications in mathematics, physics, optimization, and statistics.Invariant theory in the setting of a rational representation of a connected reductive group is themost relevant for complexity theory. The commutative case of tori is an important special case.Despite the well-understood structural simplicity of the corresponding invariant theory, even basicalgorithmic problems are non-trivial. Null cone membership, arguably the most basic problem,has long been known to have an efficient algorithm, as it reduces to linear programming, whichnon-trivially admits polynomial-time algorithms. The problems of orbit equality, orbit closureintersection, and orbit closure containment have polynomial time algorithms, as shown in this paper.We stress that while efficient algorithms for linear programming are “continuous” or “analytic” innature, our algorithms use a combination of both analytic and algebraic techniques. The moregeneral problem of succinct circuits for generating polynomial invariants, which is one of the basicchallenges proposed in [Mul17], has recently shown to be impossible under natural complexityassumptions [GIM + rational invariants for torus actions can be captured in a computationally efficient way without the needfor succinct circuits. It is an interesting open problem to determine if there are succinct circuitsfor separating invariants or null cone definers, see [GIM +
20, Problems 1.14, 1.15].The invariant theory of non-commutative groups has a different flavor from, and is far morecomplex than, the commutative case, see, for example, [Hum75]. Many interesting problems incomputational invariant theory remain open in the non-commutative case. We list a few. Firstand foremost, the results in this paper motivate the investigation of the computational efficiencyof systems of generating rational invariants. Further, it is natural to wonder if rational invariantscan help capture orbit closure intersection and orbit equality for non-commutative group actions.Another open problem is to give any polynomial time algorithm for orbit closure intersection (andthe subproblem of null cone membership). An intermediate challenge is to ascertain whether nullcone membership is in NP ∩ co-NP. Note that in [BIL +
20] it is shown that the general orbit closurecontainment problem is NP-hard.
Acknowledgements.
Peter B¨urgisser and M. Levent Do˘gan were supported by the ERC un-der the European Union’s Horizon 2020 research and innovation programme (grant agreementno. 787840); Visu Makam was supported by the University of Melbourne and by NSF grantCCF-1900460. Michael Walter acknowledges NWO Veni grant no. 680-47-459 and NWO grantOCENW.KLEIN.267. Avi Wigderson was supported by NSF grant CCF-1900460.
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Institut f¨ur Mathematik, Technische Universit¨at Berlin
Email address : [email protected] Institut f¨ur Mathematik, Technische Universit¨at Berlin
Email address : [email protected] Institute for Advanced Study, Princeton and School of Mathematics and Statistics, University ofMelbourne
Email address : [email protected] Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, and Institutefor Logic, Language and Computation, University of Amsterdam
Email address : [email protected] Institute for Advanced Study, Princeton
Email address : [email protected]@ias.edu