Detecting tropical defects of polynomial equations

We introduce the notion of tropical defects, certiﬁcates that a system of polynomial equations is not a tropical basis, and provide two algorithms for ﬁnding them in afﬁne spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.


Introduction
The tropical variety Trop(I ) of a polynomial ideal I is the image of its algebraic variety under component-wise valuation. Tropical varieties are commonly described as combinatorial shadows of their algebraic counterparts and arise naturally in many applications throughout mathematics and beyond. Inside mathematics for example, they enable new insights into important invariants in algebraic geometry [23] or the complexity of central algorithms in linear optimization [1]. Outside mathematics, they arise as spaces of phylogenetic trees in biology [25,29], loci of indifference prizes in B Yue Ren yue.ren@mis.mpg.de https://yueren.de economics [3,31] or in the proof of the finiteness of central configurations in the 4, 5-body problem in physics [10,11].
As the image of an algebraic variety, a tropical variety equals the intersection of all tropical hypersurfaces of the polynomials inside the ideal. A natural question in this context is whether this equality already holds for a given finite generating set F ⊆ I , i.e., ( * ) We call Trop(F) a tropical prevariety and, if equality holds, F a tropical basis. This question is important for two main reasons. On the one hand, tropical prevarieties can provide upper dimension bounds where Gröbner bases are infeasible to compute, see [10,11], and a tropical basis implies that this bound is actually sharp. On the other hand, the difference between a tropical variety and prevariety can be interesting in and of itself, e.g., tropical matrices of Kapranov rank r versus tropical matrices of tropical rank r [9], tropical Grassmannians versus their Dressians [14], or other realizability loci of combinatorial objects such as -matroids [28] or gaussoids [5]. Nevertheless, checking the equality in ( * ) is a computationally highly challenging task. Current algorithms for computing tropical varieties require a Gröbner basis for each maximal Gröbner polyhedron, of which there can be many even for tropicalization of linear spaces [19]. Additionally, it is known that deciding the equality in ( * ) is co-NP-hard, as is merely deciding whether Trop(F) is connected [30].
In practice, testing the equality in ( * ) can fail for multiple reasons: (P1) Computing Trop(F) might not be possible due to its size or due to the number of intersections necessary to compute it. (P2) Computing Trop(I ) might not be feasible due to its size or due to problematic Gröbner cones in Trop(I ) whose Gröbner bases are too hard to compute.
In this article, we introduce the notion of tropical defects, certificates for generating sets which are not tropical bases, and propose two randomized algorithms for computing tropical defects around affine subspaces of complementary dimension. An independent verification of these certificates will require a single Gröbner basis computation.
The basic idea is simple, relying on some recent results on (stable) intersections of tropical varieties [18,24]: To reduce the complexity of the computations, we (stably) intersect both sides of Equation ( * ) with a random affine space of complementary dimension, and look for differences between the tropical variety and prevariety around it. Under certain genericity assumptions, this yields a zero-dimensional tropical variety on the left, which is not only simpler to compute than its positive-dimensional counterparts, but also implies that the tropical prevariety computation on the right can be aborted if a positive-dimensional polyhedron is found. Therefore, our algorithm operates within the realm where (P1) and (P2) are infeasible, but the following key computational ingredients are not: (K1) computation of zero-dimensional tropical varieties in Singular [8,15], (K2) computation of zero-dimensional tropical prevarieties in DynamicPrevariety [17].
To a degree, our approach for finding tropical defects is related to the approach for studying tropical bases in [12,13]. In [12,13], the authors consider preimages of projections to R d+1 , where d := dim Trop(I ). Our hyperplanes are generally given as preimages of points under a projection to R d , but can also be regarded as preimages of lines under a projection to R d+1 . Hence, our approach can be seen as a relaxation where instead of considering the preimage of the entire projection to R d+1 we only consider the parts of the projection which meet a fixed line.
In Sects. 3 and 4, we present two tropical defects found using out algorithm, disproving Conjecture 5.3 in [27] and Conjecture 8.4 in [5]. Note that the tropical defects were postprocessed for the ease of reproduction; see Remark 2.8. Code and auxiliary materials for this article are available at software.mis.mpg.de. More information on gaussoids can be found at gaussoids.de.

Tropical defects
In this section, we introduce the notion of tropical defects for generating sets of polynomial ideals, and two algorithms to find them around generic affine spaces L = Trop(H ) of complementary dimension. To be precise, Algorithm 2.9 requires a generic tropicalization L, whereas Algorithm 2.13 merely requires a generic realization H .
We begin by briefly recalling some basic notions of tropical geometry that are of immediate relevance to us. Our notation coincides with that of [21], to which we refer for a more in-depth introduction of the subject.

Convention 2.1
For the remainder of this article, fix an algebraically closed field K with valuation ν : K * → R and residue field K with trivial valuation. Since K is algebraically closed, there is a group homomorphism μ : ν(K * ) → K * such that ν • μ = id ν(K * ) , and we abbreviate t λ := μ(λ) for λ ∈ ν(K * ). Moreover, we fix a multivariate (Laurent) polynomial ring Definition 2.2 (Initial forms, initial ideals) Given a polynomial f ∈ K [x ±1 ], say f = α∈Z n c α · x α , its initial form with respect to a weight vector w ∈ R n is For a finite set F ⊆ K [x ±1 ] and an ideal I K [x ±1 ], we denote Moreover, the Gröbner polyhedron of f , of I or of a finite set F ⊆ K [x ±1 ] around w is defined as Note that both C w ( f ) and C w (F) are in fact convex polyhedra, while C w (I ) is only guaranteed to be a convex polyhedron if I is homogeneous.
We call a finite generating set F ⊆ I a tropical basis if Note that Trop( f ), Trop(I ) and Trop(F) are supports of polyhedral complexes. For both Trop( f ) and Trop(F) these polyhedral complexes can be chosen to be a collection of Gröbner polyhedra, and, if I is homogeneous, so can Trop(I ).
Let T ⊆ R n be the support of a polyhedral complex . Recall that the star of T around a point w ∈ R n is given by and that the stable intersection of T with respect to an affine subspace H ⊆ R n is defined to be

Example 2.4
Let K = C{{t}} be the field of complex Puiseux series and consider the ideal I K [x ±1 , y ±1 ] which can be generated by either one of the following two generating sets: Figure 1 compares the tropical prevarieties of both F 1 and F 2 with the tropical variety of I , showing that F 2 is a tropical basis while F 1 is not.
For the following result, we refer to [21], where it is only shown for polynomial rings. However, the result extends directly to Laurent polynomial rings, since In particular, for a finite set We will now introduce the notion of a tropical defect and two algorithms for finding them around affine spaces of complementary dimension. For the sake of simplicity, we restrict ourselves to affine spaces in direction of the last few coordinates; see Example 2.10 for general affine spaces.
] be a polynomial ideal with finite generating set F ⊆ I . We call a finite tuple w := (w 0 , . . . , w k ) ∈ (R n ) k+1 a tropical defect if for all ε > 0 sufficiently small we have

Remark 2.8 (Singleton tropical defects)
Note that any tropical defect (w 0 , . . . , w k ) of a homogeneous ideal can be transformed into a singleton tropical defect u through a single (tropical) Gröbner basis [6] or standard basis computation [22]: One can simulate the weight vector w ε := w 0 + εw 1 + · · · + ε k w k for ε > 0 sufficiently small through a sequence of weights as in Lemma 2.5. In particular, we can compute a Gröbner basis with respect to the sequence of weights, which gives us the inequalities and equations of the Gröbner cone C w ε (I ) by [21, proof of Prop. 2.5.2]. Any u ∈ Relint C w ε (I ) is a singleton tropical defect.
For the ease of verification, the tropical defects in Sects. 3 and 4 have been transformed into singletons.
where π : R n → R d denotes the projection onto the first d coordinates.
(2) v ∈ R d , describing an affine subspace H := π −1 (v) ⊆ R n of complementary dimension n − d such that the following strong genericity assumption holds: If the algorithm terminates at Line 5, then C w (F ) is a positive-dimensional polyhedron contained in Trop(F ) = Trop(F) ∩ H , whereas Trop(I ) ∩ H consists of finitely many points. In particular, we have that w +εu / ∈ Trop(I ) for ε > 0 sufficiently small. If the algorithm terminates at Line 8, then w is a tropical defect since and let π : R {x,y} → R {x} denote the projection onto the x-coordinate. Figure 3 shows the tropical variety Trop(I ) and the tropical prevariety Trop(F).
Then, for any v ∈ R the affine line H v := π −1 (v) satisfies (SG). Algorithm 2.9 yields a tropical defect if and only if v = 0, in which case it terminates at Line 5.
We can also use arbitrary rational affine subspaces like L v := v ·e x +Span(e x +e y ) by applying a unimodular transformation ψ on the ring of Laurent polynomials whose induced map ψ on the weight space aligns L v with the coordinate axes:

Example 2.11
Consider the generating set F of the following one-dimensional ideal: and let π : R {x,y,z} → R {x} denote the projection onto the x-coordinate. Figure 4 shows Trop(I ) as well as Trop(F). Consider the plane H v := π −1 (v) for some v ∈ R. Note that while any H v with v = 0 satisfies (SG), only H v with v > 0 yields a tropical defect in Algorithm 2.9, Line 5. One possibility to ascertain whether (SG) holds upon termination at Line 5 is to compute the Gröbner polyhedron C w (I ), if I is homogeneous. However, that requires a tropical Gröbner basis or standard basis, and hence might not be viable for large examples.
In practice, affine subspaces satisfying the strong genericity assumption induce several problems; see Remark 2.16. This is why we introduce Algorithm 2.13, which relies on a weakened genericity assumption. Note that, compared to Algorithm 2.9, Algorithm 2.13 requires the computation of Trop(in w (F)) for some w ∈ Trop(F)∩ H at Line 5. This is unproblematic, however, since in w ( f ) has fewer terms than f for all f ∈ F, so that Trop(in w ( f )) will be simpler than Trop( f ). In fact, generically in w ( f ) will be a binomial and Trop(in w ( f )) a linear space. Let v 1 , . . . , v k be a basis of Span(C u (in w F)). 8: return (true, (w, u, v 1 , . . . , v k )). 9: if ∃u ∈ Trop(in w F) with dim(C u (in w F) + H ) = n then 10: Let v 1 , . . . , v d be a basis of Span(C u (in w F)).  ∃(w, u, v 1 , . . . , v d ) ∈ such that w / ∈ Trop(I ) then 14: return (true, (w, u, v 1 , . . . , v d )). 15: else 16: return (false, 0).
Suppose the algorithm terminates at Line 14. Again, by Lemma 2.5, there exists δ > 0 such that D := {w + εu + ε 2 v 1 + · · · + ε d+1 v d | 0 < ε < δ} ⊆ Trop(F). Any infinite subset of D has affine span w + Span(C u (in w F)), which intersects H stably. We have w / ∈ Trop(I ) = Trop(I ) ∩ st H by assumption (WG), so any polyhedron on Trop(I ) around w can only have a finite intersection with D. In particular, this implies that w + εu + ε 2 v 1 + · · · + ε k+1 v k / ∈ Trop(I ) for ε > 0 sufficiently small. Remark 2.14 (Weak genericity) If Algorithm 2.13 terminates at Line 8, then the output is correct even if the input did not satisfy the weak genericity assumption (WG), since a polyhedron in Trop(F) of too large dimension was found. On the other hand, the correctness of a tropical defect output at Step 14 does depend on the assumption (WG) on the input. In order to certify the correctness of the output regardless of the validity of (WG), one needs to check that there is no sufficiently small ε > 0 such that w + εu + ε 2 v 1 + · · · + ε d+1 v d ∈ Trop I . If I is homogeneous, this can by Lemma 2.5 be achieved by certifying that the iterated initial ideal in v d · · · in v 1 in u in w I is the entire Laurent polynomial ring K[x ±1 ].

Example 2.15
Consider the generating set from Example 2.10 (see also Fig. 3): Unlike before, Algorithm 2.13 will be unable to find a tropical defect around H v even for v = 0, always terminating at Line 16. This is because without condition (SG) H 0 need not have a zero-dimensional intersection with Trop(I ), so that its positivedimensional intersection with Trop(F) need not arise from a tropical defect.
However, Algorithm 2.13 will still find a tropical defect for L v for v = 0, in which case it terminates at Line 14.
Remark 2.16 (Strong genericity vs. weak genericity from a practical point of view) Theoretically, it is always possible to find tropical defects for generating sets which are not tropical bases using Algorithm 2.9 with the right choice of an affine subspace. In practice, however, it is much more reasonable to use Algorithm 2.13 instead. This is because generic v ∈ R d for Algorithm 2.9 usually entail high exponents in the polynomial computations, whereas generic λ ∈ (K * ) d for Algorithm 2.13 only entail big coefficients, and most computer algebra software systems such as Macaulay2 or Singular are better equipped to deal with the latter. For instance, our Singular experiments using Algorithm 2.9 regularly failed due to exponent overflows, since exponents in Singular are stored in the C ++ type signed short (bounded by 2 15 for most CPU architectures), while coefficients are stored with arbitrary precision. Remark 2.17 (Comparison with existing techniques) As hinted in the introduction, tropical basis verification is a problem that has been studied by many people. However, the only software currently capable of this task is gfan [16], which, for example, has been used to prove that the 4 × 4-minors of a 5 × n matrix form a tropical basis [7]. Its command gfan_tropicalbasis computes a tropical basis, and its command gfan_tropicalintersection for computing tropical prevarieties Trop(F) has an optional argument -tropicalbasistest to test whether Trop(F) equals the tropical variety Trop(I ). Compared to the algorithms in gfan, our techniques have the following disadvantages and advantages.
Since our algorithms revolve around finding tropical defects, they are incapable to verify that a generating set is a tropical basis. As we only search around random hyperplanes of complementary dimension, we are also blind to lower-dimensional defects, i.e., if dim(Trop(I ) \ Trop(F)) < dim(Trop(I )) =: d, then the probability for a random affine hyperplane of codimension d to intersect Trop(I ) \ Trop(F) is zero. One example where our algorithms failed to return a definite answer is [28,Conjecture 4.8]. In

Application: Cox rings of cubic surfaces
Cox rings are global invariants of important classes of algebraic varieties. For example, they carry essential information about all morphisms to projective spaces and play a central role in the theory of universal torsors; see [2] for further details. In this section, we address [27,Conjecture 5.3] on Cox rings of smooth cubic surfaces, disproving it with a tropical defect. Definition 3.1 Consider six points p 1 , . . . , p 6 ∈ P 2 C in general position in the complex projective plane. Up to change of coordinates, we may assume that where d i satisfy certain genericity conditions; see [26, §6]. Blowing up P 2 C in these points results in a smooth cubic surface X := Bl p 1 ,..., p 6 P 2 C . The geometry of this surface is captured by its Cox ring where • E 1 , . . . , E 6 ⊆ X are the exceptional divisors over the points p 1 , . . . , p 6 ∈ P 2 C , • E 0 ⊆ X is the preimage of a line in P 2 C not containing p 1 , . . . , p 6 , and are the rational functions on X which vanish along each E i with multiplicity at least −a i (vanishing with negative multiplicity meaning poles of positive order).
For a smooth cubic surface X , the Cox ring Cox(X ) is a finitely generated integral domain with a natural set of 27 generators which are the rational functions on X establishing the linear equivalence of each of the 27 lines on the cubic surface X to a divisor of form i a i E i ∈ Div(X ); see [4,Theorem 3.2].
One can verify that w is a tropical defect, i.e., w lies in the tropical prevariety, since in w ( f ) is at least binomial for each trinomial generator f , and outside the tropical variety, since in w (I X ) contains the monomial E 6 F 56 G 6 .
For each projection, we then constructed affine lines L 1 , . . . , L k ⊆ R d+1 such that each maximal polyhedron of Trop(g) intersects at least one line. Their preimages π −1 L 1 , . . . , π −1 L k are then d-codimensional affine subspaces which were our samples for H .