Combinatorial Differential Algebra of x^p
CCombinatorial Differential Algebra of x p Rida Ait El Manssour and Anna-Laura Sattelberger
Abstract.
We link n -jets of the affine monomial scheme defined by x p to thestable set polytope of some perfect graph. We prove that, as p varies, the dimensionof the coordinate ring of the scheme of n -jets as a C -vector space is a polynomial ofdegree n + 1 , namely the Erhart polynomial of the stable set polytope of that graph.One main ingredient for our proof is a result of Zobnin who determined a differentialGr¨obner basis of the differential ideal generated by x p . We generalize Zobnin’s resultto the bivariate case. We study ( m, n ) -jets, a higher-dimensional analog of jets, andrelate them to regular unimodular triangulations of the m × n -rectangle. Contents
Introduction 11. Differential ideals and jets 32. Linking dim C ( R n /C p,n ) and dim C ( R m,n /C p, ( m,n ) ) to lattice polytopes 103. Regular unimodular triangulations of the m × n -rectangle 12References 15 Introduction
Differential Algebra—an infinite version of polynomial algebra in a sense—studiespolynomial partial differential equations with tools from Commutative Algebra. Differ-ential Algebraic Geometry studies varieties that are defined by a system of polynomialPDEs. An upper bound for the number of components of such a variety was recentlyconstructed in [14]. Differential Algebraic Geometry comes with an own version ofthe Nullstellensatz, the differential
Nullstellensatz, relating points of a differential va-riety with formal power series solutions of the defining system of equations. Lowerand upper bounds for the effective differential Nullstellensatz are provided in [11]. Inthis article, we transfer the combinatorial flavor of Commutative Algebra [16] to Dif-ferential Algebra and undertake first steps in
Combinatorial Differential Algebra . Wepresent a case study of the fat point x p on the affine line.Denote by C p,n the ideal in R n = C [ x , . . . , x n ] generated by the coefficients of f p,n = ( x + x t + · · · + x n t n ) p , read as polynomial in the variable t with coefficientsin R n . The affine scheme defined by C p,n is the scheme of n -jets of the fat point x p on the affine line. B. Sturmfels suggested to investigate the following question. Question 2.1.
For fixed n ∈ N , is the sequence (dim C ( R n /C p,n )) p ∈ N a polynomialin p of degree n + 1 ? Mathematics Subject Classification.
Key words and phrases.
Differential ideal, Gr¨obner basis, multivariate jet, perfect graph, regulartriangulation, lattice polytope. a r X i v : . [ m a t h . AG ] F e b RIDA AIT EL MANSSOUR AND ANNA-LAURA SATTELBERGER
The point of departure are experimental observations. A first main result of thisarticle is the proof that this question has a positive answer.One main tool for the proof is a result from Differential Algebra. The object ofstudy is the differential ring C [ x ( ∞ ) ] = ( C [ x, x (1) , x (2) , . . . ] , ∂ ) , i.e., the polynomial ringin the countably infinitely many variables { x ( k ) } k ∈ N with the differential ∂ actingas ∂ ( x ( k ) ) = x ( k +1) and ∂ | C ≡ . An ideal I in C [ x ( ∞ ) ] is a differential ideal if ∂ ( I ) ⊆ I. Zobnin [24] proved that the singleton { x p } is a differential Gr¨obner basis ofthe differential ideal generated by x p with respect to any β -ordering. Denote by I p,n the differential ideal generated by x p and x ( n ) . Then the map R n /C p,n ∼ = −→ C [ x ( ∞ ) ] /I p,n +1 , x k (cid:55)→ k ! x ( k ) is an isomorphism and Zobnin’s result can be used to investigate C p,n . An investigationof the leading monomials of C p,n then reveals the following. Proposition 2.5. As p varies, dim C ( R n /C p,n ) is polynomial of degree n + 1 . It isthe Erhart polynomial of the convex polytope P n := (cid:8) ( u , . . . , u n ) ∈ ( R ≥ ) n +1 (cid:12)(cid:12) u i + u i +1 ≤ for all ≤ i ≤ n − (cid:9) evaluated at p − , i.e., it counts the lattice points of the polytope P n dilated by p − . A study of jet schemes of monomial ideals was also undertaken in [9]. Therein,it is shown that jet schemes of monomial ideals are in general not monomial, buttheir reduced subschemes are. A study of the multiplicity of jet schemes of simplenormal crossing divisors was undertaken by C. Yuen in [23]. In [22], she introduced truncated m -wedges , a two-dimensional analog of jets, studying differentials in twovariables whose orders add up to m at most. In Definition 1.9, we introduce anothergeneralization of jets to higher dimensions, namely ( m, n ) -jets , allowing for derivativesin the variables up to order m and n, respectively.We extend our studies of dim C ( R n /C p,n ) to the case of two independent vari-ables and give a link to regular unimodular triangulations. For the theory of tri-angulations, we refer our readers to [5, 21]. We study the partial differential ring C [ x ( ∞ , ∞ ) ] := (cid:0) C [ x ( k,(cid:96) ) ] k,(cid:96) ∈ N , ∂ s , ∂ t (cid:1) in two independent variables s, t and consider thedifferential ideal I p, ( m,n ) generated by x p , x ( m, , and x (0 ,n ) . Denote by C p, ( m,n ) theideal in C [ { x k,(cid:96) } ≤ k ≤ m, ≤ (cid:96) ≤ n ] generated by the coefficients of f p, ( m,n ) := (cid:32) m (cid:88) k =0 n (cid:88) (cid:96) =0 x k,(cid:96) t k s (cid:96) (cid:33) p , read as bivariate polynomial in s and t. We refer to the affine scheme associated to C p, ( m,n ) as the scheme of ( m, n ) -jets of x p . The ideals I p, ( m,n ) and C p, ( m,n ) then arerelated just as in the univariate case.For a triangulation T of the m × n -rectangle and fixed p, we define T -orderings on the truncated partial differential ring C [ x ( ≤ m, ≤ n ) ] as those monomial orderingsfor which the leading monomials of { ( x p ) ( k,(cid:96) ) } k =0 ,...,mp,(cid:96) =0 ,...,np are supported on thetriangles of T. Note that this is in contrast to the usual occurrence of regular trian-gulations in Combinatorial Commutative Algebra, where the leading monomials aresupported on non- faces (see for instance Sturmfels’ correspondence [5, Theorem 9.4.5]).We consider the placing triangulation T m, of the point configuration[(0 , , (0 , , (0 , , (1 , , (1 , , (1 , , . . . , ( m, , ( m, , ( m, . ombinatorial Differential Algebra of x p This is a regular unimodular triangulation of the m × , , , . . . , m +1 ) in the lower hull convention.We formulate the following conjectural generalization of Zobnin’s result to the partialdifferential ring in two independent variables. Conjecture 1.14.
For all m, p ∈ N , { ( x p ) ( k,(cid:96) ) } ≤ k ≤ mp, ≤ (cid:96) ≤ p is a Gr¨obner basis of thedifferential ideal generated by x p in the truncated partial differential ring C [ x ( ≤ m, ≤ ] w.r.t. any T m, -ordering. As pointed out in Proposition 1.15, we have computational evidence that this con-jecture holds true. This theorem is the main ingredient for the following proposition.
Proposition 2.6.
For m ≤ and p ≤ , the number dim C ( R m, /C p, ( m, ) is theErhart polynomial of the m + 1) -dimensional lattice polytope P ( m, := (cid:8) ( u , u , u . . . , u m , u m , u m ) ∈ ( R ≥ ) m +1) (cid:12)(cid:12) u k ,l + u k ,l + u k ,l ≤ for all indices s.t. { ( k , l ) , ( k , l ) , ( k , l ) } is a triangle of T m, (cid:9) evaluated at p − . In Section 3, we study regular unimodular triangulations of the m × n -rectangle. Weconsider the weighted degree reverse lexicographical ordering on C [ { x k,(cid:96) } ≤ k ≤ m, ≤ (cid:96) ≤ n ]for vectors inducing those triangulations in the upper hull convention. We show thatfor some of them, the coefficients of f p, ( m,n ) are a Gr¨obner basis of the ideal C p, ( m,n ) . We end our article with an outlook to future work. Our results suggest to furtherdevelop
Combinatorial Differential Algebra . Acknowledgments.
We thank Bernd Sturmfels for suggesting to work on the problemand for insightful discussions. We are grateful to Alex Fink and Gleb Pogudin for usefuldiscussions and to Taylor Brysiewicz, Lars Kastner, Marta Panizzut, and Paul Vaterfor their great support with computations in polymake [8] and
TOPCOM [19].1.
Differential ideals and jets
One independent variable.
In this section, we repeat basics from differentialalgebra and give a link to the theory of jet schemes. For further background ondifferential algebra, we refer the reader to the books [12, 20].Consider the polynomial ring C [ x, x (1) , x (2) , . . . ] in the countably infinitely manyvariables { x ( k ) } k ∈ N , where x := x (0) . Denote by C [ x ( ∞ ) ] the differential ring C [ x ( ∞ ) ] := (cid:16) C [ x, x (1) , x (2) , . . . ] , ∂ (cid:17) , where, denoting x = x (0) , the differential is given as ∂ ( x ( k ) ) = x ( k +1) and ∂ | C ≡ . Definition 1.1.
An ideal
I (cid:47) C [ x ( ∞ ) ] is called differential ideal if ∂ ( I ) ⊆ I. For asubset J of C [ x ( ∞ ) ] , (cid:104) J (cid:105) ( ∞ ) denotes the differential ideal generated by J. We denote by I p,n := (cid:104) x p , x ( n ) (cid:105) ( ∞ ) the differential ideal in C [ x ( ∞ ) ] generated by x p and x ( n ) and by C [ x ( ≤ n ) ] the truncated differential ring C [ x ( ∞ ) ] / (cid:104) x ( n +1) (cid:105) ( ∞ ) . For n ∈ N , denote by R n := C [ x , . . . , x n ] RIDA AIT EL MANSSOUR AND ANNA-LAURA SATTELBERGER the polynomial ring in n + 1 variables with coefficients in the complex numbers. Con-sider f p,n = ( x + x t + · · · + x n t n ) p ∈ R n [ t ] . By the multinomial theorem, f p,n = (cid:88) k + ··· + k n = p (cid:18) pk , k , . . . , k n (cid:19) x k · · · x k n n t k +2 k + ··· + nk n , where (cid:18) pk , k , . . . , k n (cid:19) = p ! k ! · · · k n ! . Denote by C p,n (cid:47) R n the ideal generated by the coefficients of f p,n . This ideal de-fines the scheme of n -jets of the affine scheme Spec( C [ x ] / (cid:104) x p (cid:105) ) . Up to constants, thecoefficient of t k in f p,n recovers the k -th derivative of the monomial x p , giving riseto the following relation between the differential ideal I p,n and the ideal C p,n in thepolynomial ring R n . Proposition 1.2.
The following map is an isomorphism of C -algebras: R n /C p,n ∼ = −→ C [ x ( ∞ ) ] /I p,n +1 , x k (cid:55)→ k ! x ( k ) . Proof.
Notice that ( x p ) ( k ) is given as follows:( x p ) ( k ) = (cid:88) j + ··· + j p − = k (cid:18) kj , . . . , j p − (cid:19) x ( j ) · · · x ( j p − ) . Let us consider its image in the truncated differential ring C [ x ( ≤ n ) ] . We denote by i (cid:96) the multiplicity of (cid:96) in the multiset { j , . . . , j p − } , so that i + · · · + i n = p and i + 2 i + · · · + ni n = k. Let y i := x ( i ) for all 0 ≤ i ≤ n. Then( x p ) ( k ) = (cid:88) j + ··· + j p − = k (cid:18) kj , . . . , j p − (cid:19) y i · · · y i n n . In the previous sum, there are some repeated terms: for each { j , . . . , j p − } by ex-changing the order of j i and respecting the numbers i , . . . , i n , we get the same term.We have (cid:0) pi (cid:1) possibilities to choose i many places for 0 in the multiset { j , . . . j p − } . We have (cid:0) p − i i (cid:1) possibilities to choose i many places for 1 from the remaining placesin the set { j , . . . , j p − } . We continue like this and obtain( x p ) ( k ) = (cid:88) ( i ,...,i n ) ∈ I (cid:18) pi , . . . , i n (cid:19) · k !(0!) i · · · ( n !) i n · y i · · · y i n n , where I = { ( i , i , . . . , i p ) | i + · · · + i n = p and i + · · · + ni n = k } . Denote by ϕ thefollowing homomorphism of rings: ϕ : C [ x ( ≤ n ) ] → R n /C p,n , x ( k ) (cid:55)→ k ! · x k . This homomorphism maps ( x p ) ( k ) to the coefficient of t k in the polynomial f p,n mul-tiplied by k ! . The kernel of ϕ is the ideal generated by { ( x p ) ( k ) } k ∈ N . Thus, C [ x ( ∞ ) ] /I p,n +1 ∼ = C [ x ( ≤ n ) ] / (cid:104){ ( x p ) ( k ) | k ∈ N }(cid:105) ∼ = R n /C p,n , concluding the proof. (cid:3) Remark 1.3.
Proposition 1.2 follows from [17, Proposition 5.12] applied to the idealgenerated by x p . To make this article self-contained, we decided to provide a proof. (cid:52) ombinatorial Differential Algebra of x p Following [18, 24], we now repeat the concept of differential Gr¨obner bases . For that,the monomial orderings have to be compatible with ∂ in the following sense. Definition 1.4.
A monomial ordering ≺ on C [ x ( ∞ ) ] is called admissible if it satisfiesthe following properties for all monomials M , M , and M :(i) 1 ≺ M if M (cid:54) = 1 . (ii) M ≺ M implies M M ≺ M M . (iii) M ≺ lm( ∂M ) if M (cid:54) = 1 . (iv) M ≺ M implies lm( ∂M ) ≺ lm( ∂M ) . Example 1.5.
The degrevlex ordering is an admissible ordering. We order the vari-ables as x < x (1) < x (2) < . . . If M = x i m m · · · x i n n , where m = min { k | i k (cid:54) = 0 } and x k is identified with x ( k ) , then lm( ∂M ) = x i m − m x i m +1 +1 m +1 · · · x i n n , which implies M ≺ lm( ∂M ) . If M = x i m m · · · x i n n ≺ M = x i m (cid:48) m (cid:48) · · · x i n (cid:48) n (cid:48) , where m = min { k | i k (cid:54) = 0 } and m (cid:48) = min { k | i (cid:48) k (cid:54) = 0 } , then lm( ∂M ) = x i m − m x i m +1 +1 m +1 · · · x i n n ≺ lm( ∂M ) = x i m (cid:48) − m (cid:48) x i m (cid:48) +1 +1 m (cid:48) +1 · · · x i n (cid:48) n (cid:48) . (cid:52) Definition 1.6.
Fix an admissible monomial ordering ≺ on C [ x ( ∞ ) ] and let I (cid:47) C [ x ( ∞ ) ] be a differential ideal. A subset of polynomials G ⊆ I s.t. (cid:104) G (cid:105) ( ∞ ) = I is a differential Gr¨obner basis of I if { ∂ k ( g ) | k ∈ N , g ∈ G } is an algebraic Gr¨obner basisof I (cid:47) C [ x, x (1) , x (2) , . . . ] w.r.t. ≺ . Zobnin studied the differential ideal (cid:104) x p (cid:105) ∞ and proved the following. Theorem 1.7 ([24]) . The singleton { x p } is a differential Gr¨obner basis of (cid:104) x p (cid:105) ( ∞ ) w.r.t. the reverse lexicographical ordering. Remark 1.8.
Zobnin proved this result for so called β -orderings , i.e., monomial order-ings on C [ x ( ∞ ) ] for which the leading monomial of ( x p ) ( k ) is of the form( x ( i ) ) a ( x ( i +1) ) p − a (see [13]). Since, in this article, we do not need the statementin its full generality, we just point out that the reverse lexicographical ordering issuch a β -ordering. Note moreover that ( x p ) ( k ) is bihomogeneous w.r.t. the vectors(1 , , , . . . ) and (0 , , , , . . . ) , i.e., every monomial summand (cid:81) i ( x ( i ) ) a i in ( x p ) ( k ) verifies (cid:80) i a i = p, and (cid:80) i ia i = k. (cid:52) Two independent variables.
In this subsection, we generalize Proposition 1.2to two independent variables. We denote by C [ x ( ∞ , ∞ ) ] the partial differential ring C [ x ( ∞ , ∞ ) ] := (cid:16) C [ x ( k,(cid:96) ) ] k,(cid:96) ∈ N , ∂ s , ∂ t (cid:17) in the two independent variables s, t and the commuting differentials ∂ s , ∂ t acting as ∂ s ( x ( k,(cid:96) ) ) = x ( k +1 ,(cid:96) ) , ∂ t ( x ( k,(cid:96) ) ) = x ( k,(cid:96) +1) , ∂ s | C ≡ ∂ t | C ≡ . For m, n ∈ N , denote by I p, ( m,n ) the differential ideal (cid:104) x p , x ( m, , x (0 ,n ) (cid:105) ( ∞ , ∞ ) in C [ x ( ∞ , ∞ ) ] generated by x p , x ( m, , and x (0 ,n ) . Denote by R m,n the polynomial ring in the ( m + 1)( n + 1) many variables { x k,(cid:96) } ≤ k ≤ m, ≤ (cid:96) ≤ n and let f p, ( m,n ) be the bivariate polynomial f p, ( m,n ) := (cid:32) m (cid:88) k =0 n (cid:88) (cid:96) =0 x k,(cid:96) t k s (cid:96) (cid:33) p ∈ R m,n [ s, t ] . RIDA AIT EL MANSSOUR AND ANNA-LAURA SATTELBERGER
By the multinomial theorem, f p, ( m,n ) = (cid:88) (cid:80) i k,(cid:96) = p (cid:18) pi , , . . . , i m,n (cid:19) · (cid:89) k,(cid:96) x i k,(cid:96) k,(cid:96) s k · i k,(cid:96) · t (cid:96) · i k,(cid:96) , where ( k, (cid:96) ) ∈ { , . . . , m } × { , . . . n } and i k,(cid:96) ∈ N for all ( k, (cid:96) ) . Let C p, ( m,n ) (cid:47) R m,n denote the ideal generated by the the coefficients of f p, ( m,n ) . Definition 1.9.
We refer to Spec( R m,n /C p, ( m,n ) ) as the scheme of ( m, n ) -jets of theaffine monomial scheme defined by x p . Proposition 1.10.
The following map is an isomorphism of C -algebras: R m,n /C p, ( m,n ) ∼ = −→ C [ x ( ∞ , ∞ ) ] /I p, ( m +1 ,n +1) , x k,(cid:96) (cid:55)→ k ! (cid:96) ! · x ( k,(cid:96) ) . Proof.
By the multinomial theorem, f p, ( m,n ) = (cid:88) (cid:80) i k,(cid:96) = p (cid:18) pi , , . . . , i m,n (cid:19) · (cid:89) k,(cid:96) x i k,(cid:96) k,(cid:96) s k · i k,(cid:96) · t (cid:96) · i k,(cid:96) . The coefficient f a,b of s a t b in f p, ( m,n ) is given as f a,b = (cid:88) ( i k,(cid:96) ) ∈ I (cid:18) pi , , . . . , i m,n (cid:19) · (cid:89) k,(cid:96) x i k,(cid:96) k,(cid:96) , where I = { ( i k,(cid:96) ) k,(cid:96) | (cid:80) mk =0 ( k (cid:80) n(cid:96) =0 i k,(cid:96) ) = a, (cid:80) n(cid:96) =0 ( (cid:96) (cid:80) mk =0 i k,(cid:96) ) = b, (cid:80) i k,(cid:96) = p } . Bythe symmetry of the second derivatives, we obtain ( x p ) ( a,b ) = (cid:16) ( x p ) ( a, (cid:17) (0 ,b ) = (cid:88) (cid:80) mi =0 k i = p,k +2 k + ··· mk m = a (cid:18) pk , . . . , k m (cid:19) a !(0!) k · · · ( m !) k m · ( x (0 , ) k · · · ( x ( m, ) k m (0 ,b ) = (cid:88) k ,...,k m (cid:18) pk , . . . , k m (cid:19) a !(0!) k · · · ( m !) k m (cid:88) (cid:96) + ··· + (cid:96) p − = b (cid:18) b(cid:96) , . . . , (cid:96) p − (cid:19)(cid:89) ≤ i ≤ m x ( i,(cid:96) k ... + ki − ) · · · x ( i,(cid:96) k ... + ki − ) . For all 0 ≤ i ≤ m and 0 ≤ s ≤ n, let j si be the multiplicity of s in the multi-set { l k + ··· + k i − , . . . , l k + ··· + k i − } . Thus k i = (cid:80) ns =0 j si , (cid:80) i,s j si = p, (cid:80) i,s ij si = a, and (cid:80) i,s sj si = b. Let J denote the set of all those ( j si ) s,i . Then ( x p ) ( a,b ) equals (cid:88) ( j si ) ∈ J p ! k ! · · · k m ! a !(0!) k · · · ( m !) k m b !(0!) (cid:80) j i · · · ( n !) (cid:80) j ni k ! j ! · · · j n ! · · · k m ! j m ! · · · j nm ! (cid:89) i,s (cid:16) x ( i,s ) (cid:17) j si = (cid:88) ( j si ) ∈ J (cid:18) pi , · · · i m,n (cid:19) a ! b ! (cid:89) i,s (cid:0) x ( i,s ) (cid:1) j si i ! s ! , concluding the proof. (cid:3) In order to generalize Theorem 1.7 to partial differential rings, we first generalizethe concept of β -orderings. We denote by C [ x ( ≤ m, ≤ n ) ] the truncated differential ring C [ x ( ∞ , ∞ ] / (cid:104) x ( m, , x (0 ,n ) (cid:105) ( ∞ , ∞ ) . ombinatorial Differential Algebra of x p (0 ,
0) (1 ,
0) (2 , , ,
2) (1 ,
2) (2 ,
2) (3 , · · · ( m, m, , · · · ( m, Figure 1.
The placing triangulation T m, of the m × , , (0 , , (0 , , (1 , , (1 , , (1 , , . . . , ( m, , ( m, , ( m, . Definition 1.11.
Fix m, n, p ∈ N and a triangulation T of the m × n -rectangle. Amonomial ordering ≺ on C [ x ( ≤ m, ≤ n ) ] is a T -ordering if the leading monomial of each( x p ) ( k,(cid:96) ) , ≤ k ≤ mp, ≤ (cid:96) ≤ np, is supported on a triangle of T. Remark 1.12.
By identifying x ( k,(cid:96) ) with k ! (cid:96) ! x k,(cid:96) , one equivalently defines a T -ordering as a monomial ordering on R m,n = C [ { x k,(cid:96) } ≤ k ≤ m, ≤ (cid:96) ≤ n ] s.t. the leading monomial ofeach coefficient of f p, ( m,n ) ∈ R m,n [ s, t ] is supported on a triangle of T. (cid:52) Denote by T m, the unimodular triangulation of the m × , , (0 , , (0 , , (1 , , (1 , , (1 , , . . . , ( m, , ( m, , ( m, . Note that the vector (1 , , , . . . , m +1 ) induces the triangulation T m, in the lower hull convention, hence T m, is a regular triangulation. Denote by T m,n the placingtriangulation of [(0 , , . . . , (0 , n ) , (1 , , . . . , (1 , n ) , . . . , ( m, , . . . , ( m, n )] . It consists of m copies of the triangulation in Figure 2. (0 ,
0) (1 , , , , n ) (1 , n )...(1 , , Figure 2.
The regular placing triangulation T ,n of the 1 × n -rectangle for the point configuration[(0 , , (0 , , . . . , (0 , n ) , (1 , , . . . , (1 , n )] Proposition 1.13.
For all ≤ k ≤ mp, and ≤ (cid:96) ≤ np, ( x p ) ( k,(cid:96) ) has a uniquemonomial summand supported on a triangle of T m,n . Moreover, the reverse lexicograph-ical ordering ≺ on C [ x (0 , , x (0 , , . . . , x (0 ,n ) , . . . , x ( m, , . . . , x ( m,n ) ] is a T -ordering for T = T m,n for all p, where we order the variables as x (0 , < x (0 , < · · · < x ( m,n ) . Proof.
Consider ( x p ) ( k,(cid:96) ) and let us suppose that it has a monomial summand sup-ported on a triangle of T of the form x ah,n x bh +1 ,s x ch +1 ,s +1 . Suppose that there exists amonomial M = (cid:81) j x i j, j, · · · x i j,n j,n in ( x p ) ( k,(cid:96) ) such that x ah,n x bh +1 ,s x ch +1 ,s +1 ≺ M. Sinceall monomial summands in ( x p ) ( k,(cid:96) ) have the same degree, it follows that i h,n ≤ a,i h, = · · · = i h,n − = 0 , and i j, = · · · = i j,n = 0 for all j < h. Moreover, the following
RIDA AIT EL MANSSOUR AND ANNA-LAURA SATTELBERGER identities hold: a + b + c = (cid:88) j ≥ h i j, + · · · + i j,n = p,ha + ( h + 1) b + ( h + 1) c = (cid:88) j ≥ h j ( i j, + · · · + i j,n ) = k,na + sb + ( s + 1) c = (cid:88) j ≥ h i j, + · · · + ni j,n = (cid:96). (1)Then from the second line in (1), we obtain ( a − i h,n ) + ( h + 1) (cid:0) (cid:88) j ≥ h ( i j, + · · · + i j,n ) − p (cid:1) + (cid:88) j ≥ h +2 ( j − h − i j, + · · · + i j,n ) = 0 . Thus M = x i h,n h,n x i h +1 , h +1 , · · · x i h +1 ,n h +1 ,n , i h,n = a, and i h +1 , + . . . + i h +1 ,n = b + c. Since x ah,n x bh +1 ,s x ch +1 ,s +1 ≺ M, we have i h +1 ,s ≤ b, and for all r < s, i h +1 ,r = 0 . Then fromthe third equality we have s ( b + c ) + c = s ( i h +1 ,s + · · · + i h +1 ,n ) + i h +1 ,s +1 + · · · + ( n − s ) i h +1 ,n . Thus b = i h +1 ,s and c = i h +1 ,s +1 . We conclude that M = x ah,n x bh +1 ,s x ch +1 ,s +1 . Now suppose there exists a monomial summand of ( x p ) ( k,(cid:96) ) that is supported on atriangle of T m,n of the form x ah,s x bh,s +1 x ch +1 , and suppose that there exists a monomial M such that x ah,s x bh,s +1 x ch +1 , ≺ M. Then i h,s ≤ a, i h,r = 0 for all r < s, and a + b + c = (cid:88) j ≥ h i j, + · · · + i j,n = p,ha + hb + ( h + 1) c = (cid:88) j ≥ h j ( i j, + · · · + i j,n ) = k,sa + ( s + 1) b = (cid:88) j ≥ h i j, + · · · + ni j,n = (cid:96). (2)Suppose a + b < i h,s + · · · + i h,n . Then b < i h,s +1 + · · · + i h,n . By the third line in (2),(3) s ( a + b ) + b = s ( i h,s + · · · + i h,n ) + ( i h,s +1 + · · · + ( n − s ) i h,n ) + (cid:88) j ≥ h +1 i j, + · · · + ni j,n , which is a contradiction to our assumption. From the second line in (2) we then obtain ( a + b − ( i h,s + · · · + i h,n )) + ( h + 1) (cid:88) j ≥ h ( i j, + · · · + i j,n ) − p + (cid:88) j ≥ h +2 ( j − h − i j, + · · · + i j,n ) = 0 . Thus a + b = i h,s + · · · + i h,n , and c = i h +1 , + · · · + i h +1 ,n . Therefore, from (3) weconclude that a = i h,s , b = i h,s +1 , and c = i h +1 , which means M = x ah,s x bh,s +1 x ch +1 , . We proved that if ( x p ) ( k,(cid:96) ) contains a monomial summand supported on a triangleof T m,n , then this monomial is its leading monomial. Therefore, for every 0 ≤ k ≤ mp, ≤ (cid:96) ≤ np, ( x p ) ( k,(cid:96) ) has at most one monomial summand that is supported on atriangle of T m,n . The triangles of T m,n are given by { ( j, n ) , ( j + 1 , s ) , ( j + 1 , s + 1) } and { ( j + 1 , , ( j, s ) , ( j, s + 1) } for 0 ≤ j ≤ m − s = 0 , . . . , n − . The number ofmonomials of degree p that are supported on these triangles is ( mp +1)( np +1) . Indeed,we have 2 nm triangles, (3 n + 1) m + n edges, and ( n + 1)( m + 1) vertices on T m,n . Thenumber of monomials which can be formed by the 2 nm triangles containing all threecorresponding variables is 2 nm · { a + b + c = p | a, b, c > } = 2 nm ( p − p − . The ombinatorial Differential Algebra of x p edges give rise to ((3 n + 1) m + n )( p −
1) monomials of degree p in which both variablesappear. The vertices give rise to ( n + 1)( m + 1) monomials of degree p containing onlythis variable. Then we have 2 nm ( p − p − + ((3 n + 1) m + n )( p −
1) + ( n + 1)( m + 1) =( np + 1)( mp + 1) . Each of these monomials belongs to the monomials appearing in theexpression of ( x p ) ( k,(cid:96) ) for some 0 ≤ k ≤ mp and 0 ≤ (cid:96) ≤ np. We conclude that every( x p ) ( k,(cid:96) ) has exactly one monomial that is supported on a triangle of T m,n and thismonomial is its leading monomial. (cid:3) We formulate the following conjectural generalization of Zobnin’s result to the caseof two independent variables.
Conjecture 1.14.
For all m, p ∈ N , { ( x p ) ( k,(cid:96) ) } ≤ k ≤ mp, ≤ (cid:96) ≤ p is a Gr¨obner basis of thedifferential ideal generated by x p in the truncated partial differential ring C [ x ( ≤ m, ≤ ]w.r.t. any T m, -ordering.As indicated in the following proposition, we have computational evidence that thisconjecture holds true. Proposition 1.15.
For m ≤ and p ≤ , the set of differential polynomials { ( x p ) ( k,(cid:96) ) } ≤ k ≤ mp, ≤ (cid:96) ≤ p is a Gr¨obner basis of the differential ideal generated by x p in the truncated partial differential ring C [ x ( ≤ m, ≤ ] w.r.t. any T m, -ordering.Proof. Computations in
Singular for the degrevlex ordering prove the claim for m and p as indicated in the following table:m 1 2 3 4 5 6 7 8 9 10 11 12p ≤
62 21 12 9 8 7 6 6 6 5 5 5 (cid:3)
Theorem 1.16. If m ≥ , n ≥ , and p ≥ , then the family { ( x p ) ( k,(cid:96) ) } ≤ k ≤ mp, ≤ (cid:96) ≤ np is not a Gr¨obner basis of the differential ideal generated by x p in the ring C [ x ( ≤ m, ≤ n ) ] w.r.t any T m,n -ordering.Proof. If { ( x p ) ( k,(cid:96) ) } ≤ k ≤ mp, ≤ (cid:96) ≤ np is a Gr¨obner basis of the differential ideal gener-ated by x p for the T m,n -ordering ≺ , then the same statement holds for the T m − ,n -ordering ≺ . Therefore, we restrict our proof to the case m = 1 . Let us consider the dif-ferential polynomials ( x p ) ( p − , and ( x p ) ( p − , . We will show that their S -polynomialdoes not have an LCM-representation. By [4, Theorem 2.9.6], the ( x p ) ( k,(cid:96) ) then are nota Gr¨obner basis. Note that lm(( x p ) ( p − , ) = x , x p − , and lm(( x p ) ( p − , ) = x , x p − , . Their least common multiple isLCM(lm(( x p ) ( p − , ) , lm(( x p ) ( p − , )) = x , x , x p − , . We proceed by proof by contradiction. Suppose that S (( x p ) ( p − , , ( x p ) ( p − , ) = (cid:88) a,b ( x p ) ( a,b ) g a,b , where lm(( x p ) ( a,b ) g a,b ) ≺ x , x , x p − , . Since all monomials in S (( x p ) ( p − , , ( x p ) ( p − , )are of degree p + 1 and homogeneous with respect to both derivatives ∂ s and ∂ t , wecan write S (( x p ) ( p − , , ( x p ) ( p − , ) = (cid:88) p − ≤ a ≤ p − , ≤ b ≤ c a,b ( x p ) ( a,b ) x p − − a, − b , where c a,b are constants and ( x p ) ( a,b ) x p − − a, − b ≺ x , x , x p − , . We now list the poly-nomials that can show up in the previous equality with their leading monomials:( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , , ( x p ) ( p − , x , , lm(( x p ) ( p − , x , ) = x , x p − , x , . Within all these polynomials, only ( x p ) ( p − , x , and ( x p ) ( p − , x , have leadingmonomials ≺ x , x , x p − , . Thus,(4) S (( x p ) ( p − , , ( x p ) ( p − , ) = c p − , ( x p ) ( p − , x , + c p − , ( x p ) ( p − , x , . Note that x , x p − , x , is a monomial summand of the polynomial ( x p ) ( p − , . Then x , x , x p − , x , shows up in S (( x p ) ( p − , , ( x p ) ( p − , ) but not in c p − , ( x p ) ( p − , x , + c p − , ( x p ) ( p − , x , , which is in contradiction to Equation (4). (cid:3) Linking dim C ( R n /C p,n ) and dim C ( R m,n /C p, ( m,n ) ) to lattice polytopes We now investigate the sequences dim C ( R n /C p,n ) and dim C ( R m,n /C p, ( m,n ) ) , bothconsidered as sequence in p. We link them to lattice polytopes.2.1.
Polynomiality of dim C ( R n /C p,n ) . We investigate the following question.
Question 2.1.
Fix n ∈ N . As p varies, is (dim C ( R n /C p,n )) p ∈ N a polynomial in p ofdegree n + 1 ? Before turning to the proof that this question has a positive answer, we present anexplicit example.
Example 2.2 ( dim C ( R /C p,n ) p ∈ N ) . Computations in
Singular [6] reveal the first 13entries of the sequence dim C ( R /C p,n ) p ∈ N to be0 , , , , , , , , , , , , , Mathematica , we computethe interpolating polynomial on the values for p = 1 , . . . ,
20 to be17315 p + 1790 p + 53180 p + 1972 p + 1390 p + 17360 p + 1140 p, which is indeed of degree 7 = 6 + 1 . (cid:52) Let ≺ denote the reverse lexicographical ordering on R n = C [ x , . . . , x n ] . In thefollowing lemma, we determine the initial ideal of C p,n w.r.t. ≺ . The main ingredientsfor the proof are the main result in [24] and Proposition 1.2.
Lemma 2.3.
The initial ideal of C p,n with respect to ≺ is generated by the family (cid:8) x u i i x u i +1 i +1 | u i + u i +1 = p, ≤ i ≤ n − (cid:9) . ombinatorial Differential Algebra of x p Proof.
Let us first prove that the leading monomials of our family of generators are x u i i x u i +1 i +1 . Let 0 ≤ k < np be of the form k = mp + ( p − a ) , where 1 ≤ a ≤ p and0 ≤ m ≤ n − . For k = np, the leading term of f k is x pn , where f k denotes thecoefficient of t k in the polynomial f p,n . We claim that the leading monomial of thepolynomial f k is x am x p − am +1 . Suppose that x i · · · x i n n (cid:31) x am x p − am +1 for some monomialsummand x i · · · x i n n in f k . This implies that i = · · · = i m − = 0 . Then i m + · · · + i n = p and mi m + · · · + ni n = mp + p − a = k. Since i m ≤ a, from( a − i m ) + ( m + 1)( i m + · · · + i n − p ) + ( i m +2 + · · · + ( n − m − i n ) = 0we conclude that i m = a, i m +1 = p − a, and i m +2 = · · · = i n = 0 . Therefore, x am x p − am +1 is indeed the leading monomial of f k . We now consider the truncated differential ringring C [ x ( ≤ n ) ] . As rings, C [ x ≤ n ] ∼ = C [ x , . . . , x n ] = R n . Then the following holds:in ≺ (cid:104){ ( x p ) ( k ) | ≤ k ≤ np }(cid:105) = (cid:104){ lm (cid:32) np (cid:88) k =0 [ r k ]( x p ) ( k ) (cid:33) (cid:12)(cid:12) r k ∈ C [ x ( ∞ ) ] (cid:105) , where [ r k ] denotes the equivalence class of r k in C [ x ≤ n ] . By Zobnin’s result,lm (cid:0)(cid:80) npk =0 ( x p ) ( k ) r k (cid:1) is contained in the ideal generated by the family of elements { lm( x p ) ( k ) } k ∈ N . Therefore, the initial ideal of (cid:104){ ( x p ) ( k ) } ≤ k ≤ np (cid:105) is generated by { lm(( x p ) ( k ) ) } ≤ k ≤ np , concluding the proof. (cid:3) Lemma 2.4.
The convex polytope P n := (cid:8) ( u , . . . , u n ) ∈ ( R ≥ ) n +1 (cid:12)(cid:12) u i + u i +1 ≤ for all ≤ i ≤ n − (cid:9) is a lattice polytope whose vertices are binary vectors with no consecutive s. Before proving the lemma, we recall some definitions from graph theory. Let G =( V, E ) be an undirected graph, where V denotes the set of vertices and E the setof edges. A clique of G is a complete subgraph of G. A graph is perfect if for everysubgraph, the chromatic number equals the clique number of that subgraph. A subset S ⊆ V of vertices is called stable if no two elements of S are adjacent. Borrowing thenotation from [10], the stable set polytope of G is the | V | -dimensional polytopeStab( G ) := conv (cid:8) χ S ∈ R V | S ⊆ V stable (cid:9) , where the incidence vectors χ S = ( χ Sv ) v ∈ V ∈ R V are defined as χ Sv := (cid:40) v ∈ S, . The fractional stable set polytope of G is defined asQStab( G ) := x ∈ R V | ≤ x ( v ) ∀ v ∈ V, (cid:88) v ∈ Q x ( v ) ≤ Q of G . Hence Stab( G ) = conv {{ , } V ∩ QStab( G ) } . Chv´atal [3, Theorem 3.1] proved thata graph G is perfect if and only if Stab( G ) = QStab( G ) . If follows from Fulkerson’stheory of anti-blocking polyhedra [7] that this result is equivalent to the perfect graphtheorem. The latter was conjectured by Berge [1] and proven by Lov´asz [15].
Proof of Lemma 2.4.
Consider the graph G = ( { , , . . . , n } , { [ i, i + 1] } i =0 ,...,n − ) . Ob-serve that P n is precisely the fractional stable set polytope of G. Since G is a perfectgraph, QStab( G ) = Stab( G ) and P n has binary vertices as claimed. (cid:3) For an n -dimensional polytope P ⊆ R n with integer vertices and t ∈ N , denote by L P ( t ) := | tP ∩ Z n | the number of lattice points of the dilated polytope tP. E. Erhartproved that this number is a rational polynomial in t of degree n, i.e., there existrational numbers l P, , . . . , l P,n , s.t. L P ( t ) = l P,n t n + · · · + l P, t + l P, . The polynomial L P ∈ Q [ t ] is called the Erhart polynomial of P. Proposition 2.5.
The number dim C ( R n /C p,n ) is the Ehrhart polynomial of the poly-tope P n defined in Lemma 2.4 evaluated at p − . Proof.
From Lemma 2.3 we read that x u · · · x u n n is a standard monomial if and onlyif u i + u i +1 < p for all 0 ≤ i ≤ n − . The claim then follows from Lemma 2.4. (cid:3)
Investigation of dim C ( R m, /C p, ( m, ) . In this section, we generalize the resultsfound for R n /C p,n to two independent variables, i.e., to R m,n /C p, ( m,n ) . Proposition 2.6.
For m ≤ and p ≤ , dim C ( R m, /C p, ( m, ) is the Erhart polyno-mial of the m + 1) -dimensional lattice polytope P ( m, := (cid:8) ( u , u , u . . . , u m , u m , u m ) ∈ ( R ≥ ) m +1) (cid:12)(cid:12) u k ,l + u k ,l + u k ,l ≤ for all indices s.t. { ( k , l ) , ( k , l ) , ( k , l ) } is a triangle of T m, (cid:9) evaluated at p − . Proof.
Let G be the edge graph of the regular triangulation from Figure 1 for m = 2with 3( m + 1) vertices and 2 + 7 m edges. Since this graph is perfect and the maximalcliques are precisely the triangles of T m, , Stab( G ) = QStab( G ) = P ( m, . By Theo-rem 1.15, x u · · · x u m m is a standard monomial if and only if for all triples of indices { ( i , j ) , ( i , j ) , ( i , j ) } that are a triangle of T m, , u i ,j + u i ,j + u i ,j ≤ p − . (cid:3) In terms of integer programming, Proposition 2.6 translates as follows.
Corollary 2.7.
For m ≤ and p ≤ , dim C ( R m, /C p, ( m, ) is polynomial in p ofdegree m + 1) . It is the number of non-negative integer solutions of the m linearinequalities x i ,j + x i ,j + x i ,j ≤ p − , where { ( i , j ) , ( i , j ) , ( i , j ) } runs over the m many triangles of T m, . Regular unimodular triangulations of the m × n -rectangle We now outline how regular unimodular triangulations of the m × n -rectangle giverise to T -orderings on the truncated partial differential ring C [ x ( ≤ m, ≤ n ) ] —or, equiva-lently, on the polynomial ring C [ { x k,(cid:96) } ≤ k ≤ m, ≤ (cid:96) ≤ n ] . Example 3.1 ( m = n = 2 ) . Again, denote by C p, (2 , the ideal in R , = C [ x , x , x , x , x , x , x , x , x ]generated by the (2 p + 1) many coefficients f k,(cid:96) of s k t (cid:96) in f p, (2 , = ( x + x t + x t + x s + x st + x st + x s + x s t + x s t ) p . Let ≺ denote the weighted degrevlex ordering on R , for the weight vector w , := (cid:0) + 1 , . . . , + 1 (cid:1) − (cid:0) , , , . . . , (cid:1) ∈ N , ombinatorial Differential Algebra of x p Figure 3.
The four regular unimodular triangular regulations of the 2 × T , . Note that theyall arise from T , by rotating and flipping. i.e., putting weight 128 to x , weight 127 to x , and so on. Note that w , inducesthe triangulation T , in the upper hull convention. For p = 3 , we find that within themonomials of the f k,(cid:96) , the following 8 triples of pairwise different variables show up: { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , the indices of each of which define a triangle of T , . Computations in
Singular provethat the coefficients of f , (2 , are a Gr¨obner basis of C , (2 , (cid:47) R , w.r.t. the weighteddegrevlex ordering for w , . We checked that the same statement holds true for p ≤ T , -ordering for p ≤ . There are 64regular unimodular triangulations of the 2 × (cid:52) For m = 5 , n = 2 , we validated with Singular that the coefficients of f p, (5 , area Gr¨obner basis w.r.t the weighted degrevlex ordering ≺ for a vector inducing T , in the upper hull convention up to p = 9 , approving that ≺ is a T , -ordering for p ≤ . For the 8 × p ≤ . For greater values, eventhough computing over finite characteristics, the computations are expensive and didnot terminate within several days.
Remark 3.2 (Truncated 2 -wedges) . Let us consider the truncated 2 -wedges of x p asstudied in [22]. The placing triangulation of the point configuration[(0 , , (0 , , (0 , , (1 , , (1 , , (2 , , , , , , . Com-putations in
Singular reveal that the coefficients of ( (cid:80) k + (cid:96) ≤ x k,(cid:96) s k t (cid:96) ) are a Gr¨obnerbasis w.r.t. the weighted degrevlex ordering for (32 , , , , , . Mimicking thissetup for the triangle { (0 , , (3 , , (0 , } does not give rise to a Gr¨obner basis. (cid:52) Example 3.3 ( m = 3 , n = 2 ) . Again, denote by C p, (3 , the ideal in R , = C [ x , x , x , x , x , x , x , x , x , x , x , x ]generated by the (3 p + 1)(2 p + 1) coefficients of f p, (3 , . Let ≺ denote the weighteddegrevlex ordering on R , for the weight vector w , := (2 + 1 , . . . , + 1) − (2 , , . . . , ) ∈ N . For p = 3 , the following 12 triples of pairwise different variables show up within theleading monomials of the 70 coefficients: { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , { x , x , x } , the indices of each of which define a triangle of T , . Computations in
Singular provethat the 70 coefficients f k,(cid:96) of s k t (cid:96) in f , (3 , are indeed a Gr¨obner basis of C , (3 , (cid:47)R , w.r.t. ≺ , turning ≺ into a T , -ordering for p = 3 . Note that there are 852 regularunimodular triangulations of the 3 × (cid:52) Figure 4.
The four regular unimodular triangular regulations of the 3 × T , Question 3.4.
For which m, n, p ∈ N does there exist a regular unimodular triangu-lation T of the m × n -rectangle such that the coefficients of f p, ( m,n ) are a Gr¨obnerbasis of C p, ( m,n ) w.r.t. the weighted degree reverse lexicographical ordering for a vectorinducing that triangulation in the upper hull convention? One natural continuation of the triangulation T m, of the m × m × n consists of m copies of the triangulation of the 1 × n -rectangle that is depicted inFigure 2, namely the placing triangulation of the point configuration [(0 , , (0 , , . . . , (0 , n ) , (1 , , . . . , (1 , n ) , . . . , ( m, , . . . , ( m, n )] . We point out that this triangulation does not lead to a positive answer of Question 3.4in general. For instance, T , does not give rise to a Gr¨obner basis. Only the fourtriangulations depicted in Figure 5 do. Figure 5.
The four regular uni-modular triangular regulations ofthe 1 × For m = n = 3 , the question has a negative answer. There are in total 46 . × f , (3 , are a Gr¨obner basis of C , (3 , w.r.t. the weighted degrevlex ordering. Remark 3.5.
As pointed out in [2], there are—up to symmetries— 5941 regular uni-modular triangulations of the 3 × (cid:52) ombinatorial Differential Algebra of x p It would be intriguing to find the reason for this failure and to determine all m, n ∈ N for which Question 3.4 has a positive answer. Let us point out that this problem getscomputationally expensive quickly: for the 4 × .
170 regularunimodular triangulations, whereas for the 4 × . . . Now let T be a triangulation of the m × n -rectangle as asked for in Question 3.4. Weend this article with two questions, for both of which we have computational evidence. Question 3.6.
Are the four triangulations depicted in Figure 4, continued to the m × -triangle, all regular unimodular triangulations that give rise to a Gr¨obner basis? Question 3.7. As p varies, is dim C ( R m,n /C p, ( m,n ) ) the Ehrhart polynomial of thefractional stable set polytope of the edge graph of T and is this graph perfect? References [1] C. Berge. Sur une conjecture relative au probl`eme des codes optimaux.
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Rida Ait El Manssour, Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaf-ten, Inselstraße 22, 04103 Leipzig, Germany
Email address : [email protected] Anna-Laura Sattelberger, Max-Planck-Institut f¨ur Mathematik in den Naturwissen-schaften, Inselstraße 22, 04103 Leipzig, Germany
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