Algebraic Combinatorics (ALCO)

Shitov, Yaroslav

ALGEBRAIC COMBINATORICSYaroslav ShitovPositive tropicalizations of determinantal varieties and the Gondran–Minoux rankVolume 8, issue 5 (2025), p. 1349-1352.https://doi.org/10.5802/alco.440© The author(s), 2025.This article is licensed under theCREATIVE COMMONS ATTRIBUTION (CC-BY) 4.0 LICENSE.http://creativecommons.org/licenses/by/4.0/Algebraic Combinatorics is published by The Combinatorics Consortiumand is a member of the Centre Mersenne for Open Scientific Publishingwww.tccpublishing.org www.centre-mersenne.orge-ISSN: 2589-5486Algebraic CombinatoricsVolume 8, issue 5 (2025), p. 1349–1352https://doi.org/10.5802/alco.440Positive tropicalizations of determinantalvarieties and the Gondran–Minoux rankYaroslav ShitovAbstract We point out to a connection of an old approach of Gondran and Minoux to ranksof matrices over the algebra (R, max, +) and a recent work of Brandenburg, Loho, and Sinnon positive tropicalizations of algebraic varieties. This leads to examples of tropical varietiesand their tropical bases which are not the corresponding positive tropical generating sets. Inparticular, we show that the d × d minors of a d × n matrix of variables form a positive tropicalgenerating set if and only if either (1) d = n or (2) d ⩽ 4.1. Positive tropicalizationsA common object of tropical geometry is the field C{{t}} of Puiseux series with thecorresponding valuation, which takes the formal sump =∑α∈Qpαtα with pα ∈ Cto the minimal a for which pa ̸= 0, that is, a = val p, and such an a exists because,by the definition, all exponents α with pα ̸= 0 have a common denominator andminimal element. The positive part C+ is the set of all series p ∈ C{{t}} for whichthe coefficient pa in the leading term pata is real and positive [1, 3]. The mapping(1.1) tropC+(X) = {val x : x ∈ X ∩ (C+)n} ⊆ Rnsending any algebraic variety X ⊂ (C{{t}})n to a subset of Rn is called the positivetropicalization, where the overline denotes the Euclidean closure [1, 3].Definition 1.1 ([3, Definition 2.3]). A set of polynomials F ⊂ R[x1, . . . , xn] is calleda positive tropical generating set of an ideal I ⊂ R[x1, . . . , xn] iftropC+V (I) =⋂f∈FtropC+V (f),where V (I) ⊆ (C{{t}})n is the intersection of the zero loci of all polynomials in I.As an alternative in [3, Definition 2.3], one replaces C+ by (C+) ∩ R{{t}} in (1.1),but our results and arguments are valid in both settings. Instead, one can replace C+with the set of all nonzero series in C{{t}}, and the resulting version of Definition 1.1gives the concept of tropical bases, see [6]. Indeed, there has been a known problemthat asked one to describe the triples (d, n, r) for which the (r + 1) × (r + 1) minors ofa d×n matrix of variables form a tropical basis of the variety that they generate, thatManuscript received 23rd November 2024, revised 12th March 2025, accepted 13th May 2025.Keywords. Gondran–Minoux rank, tropical varieties.ISSN: 2589-5486 http://algebraic-combinatorics.org/Yaroslav Shitovis, the set of all d×n matrices of the rank at most r, see [5]. The study of this questionbegins in the work of Develin, Santos, Sturmfels [6], and its resolution culminates inthe article of the present author, see [9] and references therein. The authors of [3]discuss a similar question in the case of positive generating sets.Question 1.2 ([3, Question 2.8]). Is there a tropical variety T and a tropical basis Bof T such that B is not a positive tropical generating set of T ?We answer this question in the positive, and we correct a statement in Table 1in [3]. Indeed, with a reference to [3, Theorem 2.4], it is said that the (r + 1) × (r + 1)minors of a d × n matrix of variables form a positive tropical generating set, providedthat r + 1 = d ⩽ n, see [3, Table 1]. Of course, this reasoning is true if the additionalcondition d = n is imposed. However, if one has d < n, the variety of rank-deficientmatrices is not anymore cut by a single minor, and Theorem 2.4 in [3] may not beapplicable. This issue was not addressed in the correction either [4], and, in fact, itturns out that, in the case d < n, the answer depends on (d, n).Theorem 1.3. Assume d ⩽ n. The d × d minors of a d × n matrix of variables forma positive tropical generating set if and only if either (1) d = n or (2) d ⩽ 4.In particular, the set of all d×d minors is not a positive tropical generating set, fora d × n matrix of variables with 5 ⩽ d < n. However, the d × d minors give tropicalbases for d × n matrices, for all possible values with d ⩽ n, as explained in [6, 9], andhence we get a positive answer to Question 1.2.2. Gondran–Minoux rankAs it turns out, the description of the set tropC+V (f) is quite well known in the casewhen f is the determinant of an m × m matrix, and the corresponding results, statedin different notation, arise to the works of Gondran and Minoux [7].Definition 2.1 ([2, Definition 3.5]). The tropical bideterminant of a real m×m matrixA is the pair (|A+|, |A−|) defined as follows, where Am and Bm are the sets of alleven and odd permutations of {1, . . . , m}, respectively:|A+| = minσ∈Am{a1σ1 + . . . + amσm}, |A−| = minτ∈Bm{a1τ1 + . . . + amτm}.Definition 2.2 ([2, Definition 2.11]). The rows of an m × n real matrix A are calledlinearly dependent in the sense of Gondran-Minoux (or just GM-dependent) if there isa partition of {1, . . . , m} into two disjoint subsets I and J , and there are real numbers(λ1, . . . , λn) such that, for any k ∈ {1, . . . , n}, one hasmini∈I{λi + aik} = minj∈J{λj + ajk}.The old result relating the bideterminant to GM-dependence is as follows.Theorem 2.3 (Gondran–Minoux theorem [2, 7]). For an m × m real matrix A, onehas |A+| = |A−| if and only if the rows of A are GM-dependent.The Gondran–Minoux theorem is equivalent to the description of the positive trop-icalizations of the varieties of singular matrices, which can appear in Section 8 of [6],Proposition 3.1 in [3], Theorem 4.1 in [1], and Lemma 2.6 in [10].Theorem 2.4. If X is an m × m matrix of variables, then tropC+V (det X) coincideswith the set of all m × m matrices A such that |A+| = |A−|.In fact, Definition 2.2 gives a deeper description of the positive tropicalization ofthe variety of rank deficient matrices.Algebraic Combinatorics, Vol. 8 #5 (2025) 1350Positive tropicalizations of determinantal varieties and GM-rankTheorem 2.5. Let d and n be integers with 1 ⩽ d ⩽ n, and let Vdn be the varietyof all d × n matrices which have the ranks less than d over C{{t}}. For a d × n realmatrix A, if the rows of A are GM-dependent, then A ∈ tropC+(Vdn).Proof. We apply Definition 2.2. Indeed, up to scaling and permutations of rows, wecan assume that there is an index s ∈ {2, . . . , d − 1} such that the condition(2.1) min{a1k, . . . , ask} = min{as+1 k, . . . , adk}holds for all k ∈ {1, . . . , n}, and we write ı̂(k) to denote one of the indices that deliverthe minimum to the left-hand side of (2.1). We proceed with the definition of thed × n matrix B with entries in C+ as follows. Indeed, for any k ∈ {1, . . . , n}, for anyi ∈ {1, . . . , s} with i ̸= ı̂(k), and, for any j ∈ {s + 1, . . . , d}, we define(bjk = tajk ) & (bik = ε · taik ) & (bı̂k =∑j bjk −∑i bik)with ε ∈ (0, 1/d). For all q ∈ {1, . . . , d}, we get val bqk = aqk with positive leadingterms, and the row sums over {1, . . . , s} and {s + 1, . . . , d} coincide, so rk B < d. □The part (2) of Theorem 1.3 is clear from the following result. Here, we use• the bideterminantal Gondran–Minoux rank δ(A), which is the largest orderof a square submatrix M of a real matrix A with |M+| ≠ |M−|, and• the Gondran–Minoux row rank GMr(A), which is the largest cardinality of aGM-independent family of the rows of A, see [2, 8].Theorem 2.6 ([8, Theorem 4.6]). If δ(A) ⩽ 3, then GMr(A) = δ(A).Theorem 2.7. Let d, n be integers with d ∈ {1, 2, 3, 4} and d ⩽ n. Then the d × dminors of a d × n matrix of variables form a positive tropical generating set.Proof. Let A be a d × n real matrix. If δ(A) = d, then some d × d submatrix M of Ahas |M+| ̸= |M−|, and its determinant polynomial f satisfies A /∈ tropC+ V (f), dueto Theorem 2.4. If δ(A) < d, then we have A ∈ tropC+ Vdn by Theorems 2.5 and 2.6.Therefore, the family tropC+ Vdn is the intersection of all tropicalizations tropC+ V (f)in which f runs over all determinants of the d × d minors, as desired. □We complete the proof of Theorem 1.3 with a study of the matrix in [8, Exam-ple 3.1]. Namely, we present a 5 × 6 matrix A each of whose 5 × 5 submatrices canbe lifted to singular matrices over C+ with the degrees of the entries equal to thecorresponding entries of A, but A cannot be lifted to a rank-four matrix itself.Example 2.8. The matrixA =0 1 0 0 1 10 1 1 1 0 01 1 0 0 0 01 0 0 1 0 11 0 1 0 1 0satisfies δ(A) = 4, but the positive tropicalization tropC+ V56 does not contain A.Proof. The property δ(A) = 4 can be checked straightforwardly, and, due to thesymmetry of the construction, it is sufficient to check the property |M+| = |M−| forthe matrices M obtained from A by removing either the first column or last column.Also, the full details of the corresponding checking appear in [8, Example 3.1].Algebraic Combinatorics, Vol. 8 #5 (2025) 1351Yaroslav ShitovAs to the second assertion, we argue by absurdum and take a 5 × 6 lifting Bε withentries in C+ such that rk Bε ⩽ 4 and val bij → aij as ε → 0. Up to scaling, we getBε =1 0 z1 z3 0 01 0 0 0 x5 x70 0 y1 y2 y3 y40 1 x2 0 x6 00 1 0 x4 0 x8 + Cwith real xk > 0, ys > 0, zt ⩾ 0 and val cij > 0. If λ1B1 + λ2B2 + . . . + λ5B5 = 0 isa linear combination of the rows of Bε with λi ∈ C{{t}} that certifies the inequalityrk Bε ⩽ 4, then we use the scaling again and assume(2.2) min{val λ1, val λ2, val λ3, val λ4, val λ5} = 0,and the constant terms of λi are denoted as µi ∈ C. We have µ1 + µ2 = µ4 + µ5 = 0and µ1 z1 + µ4 x2 = −y1 µ3, µ1 z3 + µ5 x4 = −y2 µ3, µ2 x5 + µ4 x6 = −y3 µ3, and alsoµ2 x7 + µ5 x8 = −y4 µ3. Using the further notation (α, β, γ) = (µ1, µ4, −µ3), we getz1α + x2β = y1γ, z3α − x4β = y2γ, −x5α + x6β = y3γ, −x7α − x8β = y4γ and(2.3) (x2 x5 + z1 x6) β = (x5 y1 + z1 y3) γ, −(x4 x7 + z3 x8) β = (x7 y2 + z3 y4) γor ((x4 x7 + z3 x8) · (x5 y1 + z1 y3) + (x2 x5 + z1 x6) · (x7 y2 + z3 y4)) · γ = 0, and thisgives γ = 0. Using (2.3), we get β = 0 and hence α = 0, which implies that allcoefficients (µ1, µ2, µ3, µ4, µ5) are zero, and this is a contradiction to (2.2). □Therefore, the d × d minors of a d × n matrix are not a positive tropical generatingset if n > d ⩾ 5, which completes the proof of Theorem 1.3.Acknowledgements. I thank David Speyer for careful reading of a draft and discussion,and editors and reviewers for processing the article and a detailed review report.References[1] Abeer Al Ahmadieh, May Cai, and Josephine Yu, Real and Positive Tropicalizations of Sym-metric Determinantal Varieties, 2024, https://arxiv.org/abs/2409.17462.[2] Marianne Akian, Stéphane Gaubert, and Alexander Guterman, Linear independence over trop-ical semirings and beyond, in Tropical and idempotent mathematics, Contemp. Math., vol. 495,Amer. Math. Soc., Providence, RI, 2009, pp. 1–38.[3] Marie-Charlotte Brandenburg, Georg Loho, and Rainer Sinn, Tropical positivity and determi-nantal varieties, Algebr. Comb. 6 (2023), no. 4, 999–1040.[4] Marie-Charlotte Brandenburg, Georg Loho, and Rainer Sinn, Corrigendum to “Tropical Posi-tivity and Determinantal Varieties”, Algebraic Combinatorics 7 (2024), no. 3, 749–751.[5] Melody Chan, Anders Jensen, and Elena Rubei, The 4 × 4 minors of a 5 × n matrix are atropical basis, Linear Algebra Appl. 435 (2011), no. 7, 1598–1611.[6] Mike Develin, Francisco Santos, and Bernd Sturmfels, On the rank of a tropical matrix, inCombinatorial and computational geometry, Math. Sci. Res. Inst. Publ., vol. 52, CambridgeUniv. Press, Cambridge, 2005, pp. 213–242.[7] M. Gondran and M. Minoux, Linear Algebra in Dioids: A Survey of Recent Results, in Algebraicand Combinatorial Methods in Operations Research (R. E. Burkard, R. A. Cuninghame-Green,and U. Zimmermann, eds.), North-Holland Mathematics Studies, vol. 95, North-Holland, 1984,pp. 147–163.[8] Ya. N. Shitov, Matrices with different Gondran–Minoux and determinantal ranks over max-algebras, Journal of Mathematical Sciences 163 (2009), no. 5, 598–624.[9] Yaroslav Shitov, When do the r-by-r minors of a matrix form a tropical basis?, J. Combin.Theory Ser. A 120 (2013), no. 6, 1166–1201.[10] Cynthia Vinzant, Real radical initial ideals, J. Algebra 352 (2012), 392–407.Yaroslav Shitov, Izumrudnaya ulitsa, dom 65, kvartira 4, 129346 Moscow, RussiaE-mail : yaroslav-shitov@yandex.ruAlgebraic Combinatorics, Vol. 8 #5 (2025) 1352

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