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Computational Algebra, J. Herzog, V. Vuletescu (2003)
Commutative Algebra, Singularities and Computer Algebra
A. Conca, E. Negri, E. Gorla (2017)
Cartwright-Sturmfels ideals associated to graphs and linear spaces
A. Conca (2005)
Linear spaces, transversal polymatroids and ASL domainsJournal of Algebraic Combinatorics, 25
A. Conca, E. Negri, E. Gorla (2016)
Universal Groebner bases and Cartwright-Sturmfels idealsarXiv: Commutative Algebra
D. Bernstein, A. Zelevinsky (1993)
Combinatorics of Maximal MinorsJournal of Algebraic Combinatorics, 2
B. Sturmfels, A. Zelevinsky (1993)
Maximal Minors and Their Leading TermsAdvances in Mathematics, 98
W. Bruns, A. Conca, Claudiu Raicu, M. Varbaro (2022)
Determinants, Gröbner Bases and CohomologySpringer Monographs in Mathematics
P. Hilton (1962)
Linear SpacesNature, 195
J. Okniński (1988)
On monomial algebrasArchiv der Mathematik, 50
A. Conca, M. Tsakiris (2019)
Resolution of ideals associated to subspace arrangementsAlgebra & Number Theory
A. Conca, E. Negri, E. Gorla (2013)
Universal Groebner bases for maximal minorsarXiv: Commutative Algebra
Dustin Cartwright, B. Sturmfels (2009)
The Hilbert scheme of the diagonal in a product of projective spacesarXiv: Algebraic Geometry
A. Conca, E. Negri, E. Gorla (2016)
Multigraded Generic Initial Ideals of Determinantal IdealsarXiv: Commutative Algebra
Adam Boocher (2011)
Free Resolutions and Sparse Determinantal IdealsarXiv: Commutative Algebra
A. Conca, V. Welker (2018)
Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varietiesAlgebra & Number Theory
W. Bruns, A. Conca (2003)
Gröbner Bases and Determinantal IdealsarXiv: Commutative Algebra
M. Brion (2002)
Multiplicity-free subvarieties of flag varietiesarXiv: Algebraic Geometry
(2022)
Radical generic initial ideals, to appear in the volume in honor of Bernd Sturmfels, Vietnam
Can one tell if an ideal is radical just by looking at the degrees of the generators? In general, this is hopeless. However, there are special collections of degrees in multigraded polynomial rings, with the property that any multigraded ideal gener- ated by elements of those degrees is radical. We call such a collection of degrees a radical support. In this paper, we give a combinatorial characterization of radical supports. Our characterization is in terms of properties of cycles in an associated labelled graph. We also show that the notion of radical support is closely related to that of Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded gen- erators whose multidegrees form a radical support is a Cartwright-Sturmfels ideal. Conversely, a collection of degrees such that any multigraded ideal generated by ele- ments of those degrees is Cartwright-Sturmfels is a radical support. Keywords Radical ideals · Cartwright-Sturmfels ideals · Multigraded generic initial ideals Mathematics Subject Classification Primary 13C13 · 13C70 · Secondary 13P10. Communicated by Aron Simis. The first and the second authors were partially supported by PRIN 2020355B8Y “Squarefree Gröbner degenerations, special varieties and related topics” and by GNSAGA-INdAM. * A. Conca conca@dima.unige.it E. De Negri denegri@dima.unige.it E. Gorla elisa.gorla@unine.ch Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland Vol.:(0123456789) 1 3 São Paulo Journal of Mathematical Sciences 1 Radical supports Let n be a positive integer and m =(m , … , m )∈ ℕ . Let 1 n >0 S(m)= K[x ∣ 1 ≤ j ≤ n,1 ≤ i ≤ m ] ij j be a polynomial ring over a field K endowed with the standard ℤ -grading induced by setting deg(x )= e , where e ∈ ℤ is the j-th standard basis vector. We will ij j j denote S(m) only by S if there is no danger of confusion. Given A ⊆ [n]={1, 2, … , n} we will denote by S the ℤ -graded homogeneous component of degree e of S. An element f ∈ S will be said to have multide- i A i∈A gree A. Remark 1.1 Let � ≠ A ⊆ [n] and let f ∈ S . If f ≠ 0 then it cannot have multiple factors in its factorization into irreducible factors. Hence (f) is radical and the same obviously holds if f = 0. More generally we have: Lemma 1.2 Consider a collection A ={A , … , A } of non-empty subsets of [n] such 1 s that A ∩ A =� for all i ≠ j . Then for every m ∈ ℕ , for every field K and for every i j >0 choice of f ∈ S ,… , f ∈ S the ideal I =(f , … , f ) of S is radical. 1 A s A 1 s 1 s Proof We introduce a term order in S. Then the leading term of the nonzero f ’s are pairwise coprime and squarefree. Hence f ,… , f form a Gröbner basis with 1 s in(I) radical and I is radical as well. ◻ For generalities on Gröbner bases and the transfer of properties under Gröbner deformation we refer the reader to [4] or, for a complete picture, to the forthcoming [5]. The simple assertion of Lemma 1.2 suggests the following definition. Definition 1.3 A collection (repetitions are allowed) A ={A , … , A } of non- 1 s empty subsets of [n] is a radical support with respect to a field K if for every m =(m , … , m )∈ ℕ and for every choice of f ∈ S ,… , f ∈ S the ideal 1 n 1 A s A >0 1 s I =(f , … , f ) of S is radical. Furthermore we say that A is a radical support if it is a 1 s radical support for every field K. Our goal is to provide a combinatorial characterization of radical supports. We stress that in Definition 1.3 the required property should hold for every m ∈ ℕ . >0 For example if m =(1, 1, … ,1) then ℤ -graded ideals of S are indeed monomial ide- als and hence I is radical if - and only if - its minimal generators have squarefree degrees (independently of the field). As we will see, the characterization of radical supports is related to the notion of Cartwright-Sturmfels ideals. This notion has its roots in the work of Bernstein, Boocher, Brion, Cartwright, Conca, Sturmfels, Villarreal and Zelevisky [1–3, 1 3 São Paulo Journal of Mathematical Sciences 6, 7, 15, 16] among others. The theory of Cartwright-Sturmfels ideals has been developed in a series of papers by the authors of the present manuscript [8–12] and studied further by Conca and Welker in [13]. Here we quickly revise the defi- nition and the main features of Cartwright-Sturmfels ideals referring the readers to [8–12] for more general statements and for the proofs. 2 Cartwright–Sturmfels ideals We follow the notation of the previous section. The group G = GL (K)×⋯ × GL (K) acts on S as the group of multigraded K-algebra m m 1 n automorphisms. Let B = B (K)×⋯ × B (K) be the Borel subgroup of G, con- m m 1 n sisting of the upper triangular matrices in G. Let I be a multigraded ideal of S and let be a term order on S with x > x for all j and all i < k. ij kj If K is infinite we may define, as in the standard ℤ-graded situation, the (mul- tigraded) generic initial ideal gin(I) of I with respect to as in (g(I)) with g a “generic" element in G. It turns out that gin(I) is Borel fixed, i.e., fixed by the action of B. Radical Borel-fixed ideals play an important role in the definition of Cart- wright–Sturmfels ideals. Definition 2.1 We let B−rad(S) be the set of radical monomial ideals J of S such that for every monomial f ∈ S and i, j such that x f is a generator of J one has ij x f ∈ J for all k < i. kj The ideals in B−rad(S) are Borel-fixed. Indeed, at least if K is infinite, B−rad(S) is the set of Borel-fixed radical ideals, hence the name. Let M be a finitely generated ℤ -graded S-module and assume for simplicity that M = 0 if a ∉ ℕ . The multigraded Hilbert series HS (z , … , z ) of M has a rational expression M 1 n (z , … , z ) M 1 n HS (z , … , z )= . M 1 n (1 − z ) i=1 Here (z , … , z ) is a polynomial with coefficients in ℤ known as the K-polyno- M 1 n mial of M. The dual K-polynomial (z , … , z ) of M is defined as 1 n (z , … , z )= (1 − z , … ,1 − z ). 1 n M 1 n If n is clear from the context we will use HS (z) for HS (z , … , z ) and similarly M M 1 n will use (z) and (z). Definition 2.2 Let I be a multigraded ideal of S. Then I is Cartwright-Sturmfels if there exists J ∈ B − rad(S) such that HS (z) = HS (z). S∕I S∕J 1 3 São Paulo Journal of Mathematical Sciences We denote by CS(S) the family of Cartwright-Sturmfels ideals of S. It turns out that the ideal J that appears in Definition 2.2 is uniquely determined by (the multigraded Hilbert series of) I, see [8, Theorem 3.5]. This leads to the following characterization. Proposition 2.3 Assume that K is infinite. Then I is Cartwright-Sturmfels if and only if gin(I) ∈ B − rad(S). For every j ∈[n] we rename y the variable x and let T = K[y , … , y ] ⊆ S with j 1j 1 n n n induced (fine) ℤ -graded structure, i.e. deg y = e ∈ ℤ . Let M(T , m) be the set of j j the monomial ideals of T generated by monomials in the y’s, whose exponent vector is bounded from above by m =(m , … , m ). 1 n The two sets B−rad(S) and M(T , m) of monomial ideals are in one-to-one cor- respondence via the map (1) ∶M(T , m) → B − rad(S) that sends E ∈M(T , m) to (E)= J = pol(E) . Here pol(E) is the polarization of E and the star denotes the Alexander dual. When K is infinite, the inverse of can be defined as follows. (2) ∶ B − rad(S) →M(T , m) sends J ∈ B − rad(S) to (J)= E , wher e ∗ ∗ E ∈M(T , m) is uniquely determined by the property gin(J )= ES . Here J is the Alexander dual of J and ES is the extension of E to S. When K is finite, one can take an infinite extension F of K. The map is defined on F and (J)= E is an ideal generated by monic monomials. The same monomials −1 generate the ideal (J)∈M(T, m). Proposition 2.4 Let K be an arbitrary field. Given J ∈ B − rad(S) and E ∈M(T , m) one has that J corresponds to E in the bijective correspondence above if and only if (z)= (z). S∕J Furthermore we have Lemma 2.5 Let m =(m , … , m ), q =(q , … , q )∈ ℕ . Let K and L be fields, let 1 n 1 n >0 I ⊆ S = K[x ∶ j ∈[n] and i ≤ m ] and J ⊆ R = L[x ∶ j ∈[n] and i ≤ q ] be multi- ij j ij j graded ideals such that (z)= (z) . Then I ∈ CS(S) if and only if J ∈ CS(R). S∕I R∕J Proof Assume I ∈ CS(S) . Then there exists H ∈ B − rad(S) such that I and H have the same Hilbert series, in particular (z)= (z) . Then set r =(r , … , r ) with S∕I S∕H 1 n r = max{m , q } for all j ∈[n] and T = L[x ∶ j ∈[n] and i ≤ r ] . Then we have the j j j ij j inclusion R ⊆ T . We may consider the ideal H of T generated by the monomial gen- erators of H. Since H ∈ B − rad(S) we have H ∈ B−rad(T) . We can consider the extensions JT of J. Obviously the K-polynomial does not change under a polynomial extension and under the passage from H to H . Hence 1 3 São Paulo Journal of Mathematical Sciences (z)= (z)= (z)= (z)= (z). T∕JT R∕J S∕I S∕H T∕H Hence JT has the same Hilbert series of H , that is, JT ∈ CS(T) . It remains to prove that JT ∈ CS(T) implies J ∈ CS(R) . First assume that L is infinite. The computation of generic initial ideals commute with polynomial extensions, i.e. gin(JT) = gin(J)T . Since gin (JT)∈B − rad(T) we have that gin(J) ∈ B−rad(R) and hence J ∈ CS(R) . If L is finite then we can consider an infinite extension of L, compute the gin of J in with coefficients in the extension and consider the outcome in the original pol- ynomial ring R. In this way, repeating the argument above, we obtain an ideal in B−rad(R) with the Hilbert series of J. Hence J ∈ CS(R) in this case as well. ◻ Cartwright-Sturmfels ideals remains Cartwright-Sturmfels under arbitrary ℤ -graded linear section. This is proved in [9] under the assumption that the base field is infinite but the result holds in general as the reader can easily check. Proposition 2.6 Let L be an ideal of S that is generated by ℤ -graded linear forms. We identify R = S∕(L) with a polynomial ring with the induced ℤ -graded structure and J = I +(L)∕(L) with an ideal of R. If I ∈ CS(S) , then J ∈ CS(R). We conclude with a definition: Definition 2.7 A collection (repetitions are allowed) A ={A , … , A } of non-empty 1 s subsets of [n] is a Cartwright-Sturmfels support if for every m =(m , … , m )∈ ℕ 1 n >0 and for every choice of f ∈ S ,… , f ∈ S the ideal I =(f , … , f ) of S is 1 A s A 1 s 1 s Cartwright-Sturmfels. 3 Support for regular sequences and the graph associated to the support For later applications, we establish some simple facts. Lemma 3.1 Let A ={A , … , A } be a collection of non-empty subsets of [n]. The 1 s following are equivalent: (1) There exist f ∈ S ,… , f ∈ S such that f ,… , f is a regular sequence and 1 A s A 1 s 1 s (f , … , f ) is radical. 1 s (2) There exist f ∈ S ,… , f ∈ S such that f ,… , f is a regular sequence. 1 A s A 1 s 1 s (3) For every j ∈[n] we have {v ∶ j ∈ A } ≤ m . v j Proof That condition (1) implies condition (2) is obvious. For (2) implies (3) sup- pose there exist f ∈ S ,… , f ∈ S such f ,… , f is a regular sequence. For a 1 A s A 1 s 1 s given j ∈[n] the ideal (f ∶ j ∈ A ) is generated by regular sequence and contained v v 1 3 São Paulo Journal of Mathematical Sciences in the ideal generated by S . Hence we have that {v ∶ j ∈ A } ≤ m . Finally for (3) e v j implies (1) for every v ∈[s] and every j ∈ A we set k = {w ∶ w ≤ v and j ∈ A }. j,v w By assumption {v ∶ j ∈ A } ≤ m and hence k ≤ m . Therefore we may set v j j,v j f = x ∈ S v k ,j A j,v v j∈A for every v ∈[s] . Notice that k ≥ 1 , since j ∈ A . Moreover, x = x implies j,v v k ,j k ,i j,v i,w i = j and if v > w , then k > k . Hence f ,… , f is a set of pairwise coprime and j,v j,w 1 s squarefree monomials. In other words, f ,… , f is a (monomial) regular sequence 1 s that generates a radical ideal. ◻ Lemma 3.2 Suppose I =(f , … , f ) is a non-radical ideal of a ring R. Then for every 1 c 1 ≤ t ≤ c the ideal J =(f , … , f , xf , xf , … , xf ) of the polynomial extension 1 t−1 t t+1 c R[x] is not radical. Proof Consider on R[x] the graded structure associated to the assignment deg x = 1 and deg a = 0 for all a ∈ R . Then J is homogeneous and its homogeneous compo- nents J with i > 0 are all equal to I. Let a ∈ R be such that a ∉ I and a ∈ I . Then ax ∉ J and (ax) ∈ J so that J is not radical. ◻ Lemma 3.3 If A ={A , … , A } is a radical support with respect to a field K then 1 s every subcollection of A is a radical support with respect to K. Proof It suffices to observe that in Definition 1.3 some of the polynomials f ∈ S i A can be taken equal to 0. ◻ Given a collection A ={A , … , A } of subsets of [n] we define a non-oriented 1 s graph G(A) with possibly multiple edges and no loops as follows. The vertices of G(A) are labelled by elements in [s]={1, 2, … , s} and we put an edge labelled with j ∈[n] between vertex v and vertex w if j ∈ A ∩ A . This will be denoted by v w vw A cycle in G(A) is a sequence of vertices and edges of G(A) 1 3 São Paulo Journal of Mathematical Sciences j j k 1 v v k 2 with v ,… , v distinct. 1 k A k-cycle is a cycle which involves k edges and k distinct vertices. We are interested in cycles with non-constant edge labels, i.e., cycles with edge labels j ,… , j such that j ≠ j for at least a pair a, b in [k]. We observe the following. 1 k a b Lemma 3.4 If G(A) has a cycle with non-constant edge labels, then it has a cycle where all the edge labels are distinct. Proof By assumption there is a cycle, say a k-cycle, with non-constant edge labels, call it C. If k = 2 then C has distinct edge labels and we are done. So we may assume k > 2 . We may also assume that C has some repeated edge labels (otherwise, we are done). Say the edge label 1 appears more than once in C. Since the edge labels are not constant, the edge label 1 must be adjacent at least once to a different edge label, say 2. Up to a “rotation" we may hence assume that C looks like this: v v k 2 We know that there is at least another label equal to 1, say from v to v , so that p p+1 1 3 São Paulo Journal of Mathematical Sciences v v k 2 p+1 with 3 ≤ p ≤ k and p + 1 = 1 if p = k . Then 1 ∈ A ∩ A , hence the edge v v 2 p v v 2 p is in G(A) . Therefore we have a shorter cycle in G(A) with non-constant edge labels. Iterating the process, we produce a cycle where all the edge labels are distinct. ◻ Lemma 3.5 Let p ∈ ℕ with p ≥ 2 and K be any field. Set S = K[x , … , x , y , … , y ] 1 p 1 p with ℤ -graded structure given by deg x = deg y = e for every i = 1,… , p . Let i i i I =(x y − x y ∶ i = 1, 2, … , p − 1)+(y y ). i+1 i i i+1 1 p Then I is not radical. In particular, A = {{1, 2}, {2, 3},{3, 4},… , {p − 1, p}, {1, p}} is not a radical support with respect to K. Proof A straightforward application of Buchberger’s Algorithm shows that the reduced Gröbner basis of I with respect to the degree reverse lexicographic term order induced by the total order on the variables x > ⋯ > x > y > ⋯ > y 1 p 1 p 1 3 São Paulo Journal of Mathematical Sciences is obtained by adding to the given generators the monomials (x y )y , (x y )x y , (x y )x x y , … , (x y )x x ⋯ x y . 1 p 2 1 p 2 3 1 p 2 3 4 1 p 2 3 p−1 p Therefore (x y )x x ⋯ x y ∈ I and x x x ⋯ x y ∉ I . This shows that I is not 1 p 2 3 p−1 p 1 2 3 p−1 p radical, hence A is not a radical support with respect to K. ◻ 4 The main result We are ready to formulate and prove our main result. Theorem 4.1 Given a collection A ={A , … , A } of non-empty subsets of [n] we 1 s have that the following conditions are equivalent. (1) A is a radical support. (2) A is a radical support for at least one field K. (3) The graph G(A) has no cycles with distinct edge labels. (4) The graph G(A) has no cycles with non-constant edge labels. (5) There exist a field K, m =(m , … , m ) and a regular sequence f ,… , f of 1 n 1 s degrees A in S such that the ideal (f , … , f ) is Cartwright-Sturmfels. 1 s (6) For every field K, every m =(m , … , m ) such that m ≥ {v ∈[s]∶ j ∈ A } for 1 n j v all j, and every f ,… , f regular sequence of degrees A in S the ideal (f , … , f ) 1 s 1 s is Cartwright-Sturmfels. (7) A is a Cartwright-Sturmfels support. Proof It is clear that (1) implies (2). We prove that (2) implies (3) by contradiction. That is, we assume that (2) holds and that G(A) has a cycle with distinct edge labels and derive a contradiction. Renaming vertices and edges if needed, we may hence assume that G(A) contains the p-cycle p − 1 with p > 1 . By 3.3 the subcollection {A , … , A } is a radical support for K. 1 p Notice that by construction 1 3 São Paulo Journal of Mathematical Sciences A ⊇ {1, 2}, A ⊇ {2, 3}, … , A ⊇ {p − 1, p} and A ⊇ {1, p}. (4.1) 1 2 p−1 p If in (4.1) we have all equalities then by 3.5 we have polynomials f , f ,… , f of 1 2 p degree {1, 2}, {2, 3}, … , {p − 1, p}, {1, p} in a multigraded polynomial ring over K that generate a non-radical ideal and hence we have a contradiction. If instead some inclusions in (4.1) are strict we still consider the polynomials f ,… , f above and set 1 p B = (∪ A ) ⧵ {1, … , p}. i=1 For every u ∈ B we pick a new variable x of degree {u} and set f = f ( x ). i u u∈B∩A � � Now by the iterated application of 3.2 we know that the polynomials f ,… , f gen- 1 p erate a non-radical ideal and have degrees A ,… , A respectively. A contradiction. 1 p It follows from 3.4 that (3) implies (4). Equivalence of (5) and (6) follows from 2.5, since the K-polynomial of a regular sequence generated by elements of degree A is 𝕂 (z)= (1 − z )∈ ℤ[z , … , z ], A j 1 n (4.2) v=1 j∈A hence it depends only on A. Now we prove that (4) implies (5). For j ∈[n] , let m = {v ∈[s]∶ j ∈ A } . Let j v K be any field. By 3.1 there exists a regular sequence f ,… , f in S(m) of degrees 1 s A . Set I =(f , … , f ) . Then the K-polynomial of S/I is as in 4.2. In order to prove 1 s I is a Cartwright-Sturmfels ideal, by 2.4, it is suffices to exhibit a monomial ideal E in the polynomial ring T = K[y , y , … , y ] equipped with the (fine) ℤ -grading 1 2 n deg y = e ∈ ℤ such that: i i (i) the K-polynomial of E is (z) (ii) for every j, the largest exponent of y in the generators of E is bounded from above by m . We have (z)= (1 − z , … ,1 − z )= G (z , … , z ) A 1 n v 1 n v=1 with B+1 G (z , … , z )=(1 − (1 − z )) = (−1) z . v 1 n j j j∈A �≠B⊆A j∈B 1 3 São Paulo Journal of Mathematical Sciences Notice that the ideal E =(y ∶ j ∈ A ) is resolved by the truncated Koszul complex v j v ∗ n K (E ) associated to the variables y with j ∈ A and hence its ℤ -graded Hilbert v j v series is G (z , … , z ) v 1 n (1 − z ) i=1 Indeed the full Koszul complex K(E ) resolves T∕E and we take the truncated v v Koszul complex obtained from K(E ) by removing the component in homological position 0 and shifting the remaining homological positions by 1 to get a T-resolu- tion of the ideal E . We claim that, under the assumption on G(A) from (4), the ideal E = E v=1 has the properties (i) and (ii). Indeed, (ii) is satisfied by construction. We now prove (i). The minimal free resolution of the product of any collection of ideals I ,… , I 1 v generated by linear forms is described in [14]. It is proved that such a resolution is obtained as a subcomplex, supported on a specific polymatroid, of the tensor prod- uct of the truncated Koszul complexes K (I ) . We notice that assumption (4) on G(A) is equivalent to the assumption that the ideal E has �A � generators. Then v=1 the results in [14] show that the minimal free resolution of E is the tensor product ∗ ∗ ∗ K (E ) ⊗ K (E ) ⊗⋯ ⊗ K (E ) . This implies (i) and concludes the proof that (4) 1 2 v implies (5). Since every Cartwright–Sturmfels ideal is radical, then (7) implies (1). It remains to prove that (5) implies (7). Let K be any field, S = K[x ∣ 1 ≤ j ≤ n,1 ≤ i ≤ m ] be a ℤ -graded polynomial ring over K and I ij j an ideal generated by elements f ,… , f of degrees A ,… , A . For i = 1,… , s we 1 s 1 s introduce new variables t for every pair i ∈[s] and j ∈[n] such that j ∈ A , with ij i n n deg t = e ∈ ℤ . Then in the ℤ -graded polynomial ring ij j R = S[t ∶ i ∈[s] j ∈ A ] ij i we consider the polynomials g = f + t i i ij j∈A of degree A and observe that the g ,… , g form a regular sequence. Indeed, the i 1 s leading term of g with respect to the lexicographic order with the t’s larger than the x’s is t . Since the leading terms of the g ,… , g are pairwise coprime, we 1 s j∈A ij have that they form a Gröbner basis and g ,… , g form a regular sequence of ele- 1 s ments of degree A is the ℤ -graded polynomial ring R. Hence (g , … , g ) ∈ CS(R) 1 s by assumption. The g ’s specialize to to the f ’s modulo the multigraded ideal of i i linear forms (t ) . Hence I ∈ CS(S) by 2.6. ◻ ij 1 3 São Paulo Journal of Mathematical Sciences Funding Open access funding provided by Università degli Studi di Genova within the CRUI-CARE Agreement. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statu- tory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References 1. Bernstein, D., Zelevinsky, A.: Combinatorics of maximal minors. J. Algebraic Comb. 2(2), 111–121 (1993) 2. 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São Paulo Journal of Mathematical Sciences – Springer Journals
Published: Jun 1, 2023
Keywords: Radical ideals; Cartwright-Sturmfels ideals; Multigraded generic initial ideals; Primary 13C13; 13C70; Secondary 13P10.
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