Multiplicity structure of the arc space of a fat point

The equation $x^m = 0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x, x', x^{(2)}, \ldots]$ by all differential consequences of $x^m = 0$. This infinite-dimensional algebra admits a natural filtration by finite dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals $\frac{m}{1 - mt}$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by nonreduced version of the geometric motivic Poincar\'e series, multiplicities in differential algebra, and connections between arc spaces and the Rogers-Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context.

1. Introduction 1.1.Statement of the main result.Let k be a field of characteristic zero.Consider an ideal I ⊂ k[x], where x = (x 1 , . . ., x n ), defining an affine scheme X .We consider the polynomial ring k[x (∞)  ] := k[x in infinitely many variables {x ( j) i | 1 ⩽ i ⩽ n, j ⩾ 0}.This ring is equipped with a k-linear derivation a → a ′ defined on the generators by for 1 ⩽ i ⩽ n, j ⩾ 0.
Then we define the ideal ] of the arc space of X by In this paper, we will focus on the case of a fat point I m := ⟨x m ⟩ ⊂ k[x] of multiplicity m ⩾ 2. Although the zero set of I (∞) m over k consists of a single point with all the coordinates being zero, the dimension of the corresponding quotient algebra k[x (∞)  ]/I (∞)  where x (⩽ℓ) := {x, x ′ , . . ., x (ℓ) }, and arranging the dimensions of these algebras into a generating series (1) The main result of this paper is that (2) 1.2.Motivations and related results.Our motivation for studying the series (1) comes from three different areas: algebraic geometry, differential algebra, and combinatorics.
(1) From the point of view of algebraic geometry, I (∞) defines the arc space L(X ) of the scheme X [Denef and Loeser 2001].Geometrically, the points of the arc space correspond to the Taylor coefficients of the k[[t]]-points of X .The arc space of a variety can be viewed as an infinite-order generalization of the tangent bundle or the space of formal trajectories on the variety.For properties and applications of arc spaces, we refer to [Denef and Loeser 2001;Bourqui et al. 2020].
The reduced structure of an arc space is often described by means of the geometric motivic Poincaré series [Denef and Loeser 2001, §2.2] where π ℓ denotes the projection of L(X ) to the affine subspace with the coordinates x (⩽ℓ) (i.e., the truncation at order ℓ) and [Z ] denotes the class of variety Z in the Grothendieck ring [Denef and Loeser 2001, §2.3].A fundamental result about these series is the Denef-Loeser theorem [1999, Theorem 1.1] saying that P X (t) is a rational power series.
The arc spaces may also have a rich scheme (i.e., nilpotent) structure, see [Linshaw and Song 2021;Feigin and Makedonskyi 2020;Dumanski and Feigin 2023], reflecting the geometry of the original scheme [Sebag 2011;Bourqui and Haiech 2021].In the case of a fat point I m = ⟨x m ⟩ ⊂ k[x], we will have π ℓ (L(X )) ∼ = ‫ށ‬ 0 , so the geometric motivic Poincaré series is equal to where ‫ށ[‬ 0 ] is the class of a point.Note that the series does not depend on the multiplicity m of the point.One way to capture the scheme structure of L(X ) could be to take the components of the projections in (3) with their multiplicities.For example, for the case I m , one will get ∞ ℓ=0 dim k k[x (⩽ℓ)  ]/I (∞) Our result (2) implies that the series above is rational, as in the Denef-Loeser theorem.Interestingly, the shape of the denominator is different from the one in [Denef and Loeser 2001, Theorem 2.2.1].The formula above is not the only way to take the multiplicities into account.A related and more popular approach is via Arc Hilbert-Poincaré series [Mourtada 2023, §9]; see also [Mourtada 2014;Bruschek et al. 2013].
(2) Differential algebra studies, in particular, differential ideals in k[x (∞)  ], that is, ideals closed under derivation.From this point of view, I (∞) is the differential ideal generated by I .Understanding the structure of the differential ideals I (∞) m is a key component of the low power theorem [Levi 1942;1945] which provides a constructive way to detect singular solutions of algebraic differential equations in one variable.Besides that, various combinatorial properties of I (∞) m have been studied in differential algebra, see [O'Keefe 1960;Pogudin 2014;Arakawa et al. 2021;Zobnin 2005;2008;Ait El Manssour and Sattelberger 2023].
While there is a rich dimension theory for solution sets of systems of algebraic differential equations [Kondratieva et al. 1999;Pong 2006;Kolchin 1964], we are not aware of a notion of multiplicity of a solution of such a system.In particular, the existing differential analogue of the Bézout theorem [Binyamini 2017] provides only a bound, unlike the equality in classical Bézout theorem [Hartshorne 1977, Theorem 7.7, Chapter 1].Our result (2) suggests that one possibility is to define the multiplicity of a solution as the growth rate of multiplicities of its truncations, and this definition will be consistent with the usual algebraic multiplicity for the case of a fat point on a line.
(3) Connections between the multiplicity structure of the arc space of a fat point and Rogers-Ramanujan partition identities from combinatorics were pointed out by Bruschek, Mourtada, and Schepers in [2013] (for a recent survey, see [Mourtada 2023, §9]).In particular, they used Hilbert-Poincaré series of similar nature to (1) (motivated by the singularity theory [Mourtada 2014, §4]) to obtain new proofs of the Rogers-Ramanujan identities and their generalizations.In this direction, new results have been obtained recently in [Afsharijoo 2021;Afsharijoo et al. 2023;Bai et al. 2020].Afsharijoo [2021] used computational experiments to conjecture the initial ideal of I (∞) m with respect to the weighted lexicographic ordering [Afsharijoo 2021, §5] (a special case was already conjectured in [Afsharijoo and Mourtada 2020, §1]).This conjecture would imply a new set of partition identities [Afsharijoo 2021, Conjecture 5.1].Using combinatorial techniques, some of them have been proved in [Afsharijoo 2021], and the rest were established in [Afsharijoo et al. 2023]; see also [Afsharijoo et al. 2022].However, the original algebraic conjecture about I (∞)   m remained open.As a byproduct of our proof of (2), we prove this conjecture (see Theorem 3.3), thus giving a new proof of one of the main results of [Afsharijoo et al. 2023].
Understanding the structure of the ideal I (∞) m is known to be challenging: for example, its Gröbner basis with respect to the lexicographic ordering is not just infinite but even differentially infinite [Zobnin 2005;Afsharijoo and Mourtada 2020], and the question about the nilpotency index of the x Ritt [1950, Appendix, Q.5] remained open for sixty years until the paper of Pogudin [2014]; see also [O'Keefe 1960;Arakawa et al. 2021].
Statement (2) appeared in the Ph.D. thesis of Pogudin [2016, Theorem 3.4.1],but the proof given there was incorrect.We are grateful to Alexey Zobnin for pointing out the error.The proof presented in this paper uses different ideas than the erroneous proof in [Pogudin 2016].We would like to thank Ilya Dumanski for pointing out that the main dimension result (2) could also be deduced from a combination of Propositions 2.1 and 2.3 from [Feigin and Feigin 2002].
1.3.Overview of the proof.The key technical tool used in our proofs is a representation of the quotient algebra k[x (∞)  ]/I (∞) m as a subalgebra in a certain differential exterior algebra that is constructed in [Pogudin 2014]; see Section 4.1.The injectivity of this representation builds upon the knowledge of a Gröbner basis for I (∞) m with respect to the degree reverse lexicographic ordering [Bruschek et al. 2013;Zobnin 2008;Levi 1942].We approach (2) as a collection of inequalities The starting point of our proof of the lower bound uses the insightful conjecture by Afsharijoo [2021, §5] that suggests how the standard monomials of I (∞) m with respect to the lexicographic ordering look like.Using the exterior algebra representation, we prove that these monomials are indeed linearly independent modulo I (∞)  m , and deduce the lower bound from this; see Section 4.3 and 4.4.In order to prove the upper bound from (4), we represent the image of k[x (⩽ℓ)  ]/I (∞) m in the differential exterior algebra as a deformation of an algebra which splits as a direct product of ℓ + 1 algebras of dimension m, thus yielding the desired upper bound; see Section 4.2.
1.4.Structure of the paper.The rest of the paper is organized as follows: Section 2 contains definitions and notations used to state the main results.Section 3 contains the main results of the paper.The proofs of the results are given in Section 4. Then Section 5 describes computational experiments in [Macaulay2] that we performed to check whether formulas similar to (2) hold for more general fat points in k n .We formulate some open questions based on the results of these experiments.
Definition 2.1 (differential rings and fields).A differential ring (R, ′ ) is a commutative ring with a derivation ′ : R → R, that is, a map such that, for all a, b ∈ R, we have (a+b) ′ = a ′ +b ′ and (ab) ′ = a ′ b+ab ′ .A differential field is a differential ring that is a field.For i > 0, a (i) denotes the i-th order derivative of a ∈ R.
Analogously, we can define x (⩽h) .If x = (x 1 , . . ., x n ) is a tuple of elements of a differential ring, then and extend the derivation from R to this ring by (x ( j) ) ′ := x ( j+1) .The resulting differential ring is called the ring of differential polynomials in x over R. The ring of differential polynomials in several variables is defined by iterating this construction.
Definition 2.4 (differential ideals).Let S := R[x (∞)  1 , . . ., x (∞) n ] be a ring of differential polynomials over a differential ring R.An ideal I ⊂ S is called a differential ideal if a ′ ∈ I for every a ∈ I .One can verify that, for every f 1 , . . ., f s ∈ S, the ideal is a differential ideal.Moreover, this is the minimal differential ideal containing f 1 , . . ., f s , and we will denote it by ⟨ f 1 , . . ., f s ⟩ (∞) .Definition 2.5 (fair monomials).( 1 ], we define the order and lowest order, respectively, as )  ] is called fair (respectively, strongly fair) We denote the sets of all fair and strongly fair monomials by F and F s , respectively.By convention, 1 ∈ F and 1 ∈ F s .Note that F s ⊂ F.
(3) For every integers a, b ⩾ 0, we define where the product of sets of monomials is the set of pairwise products.In other words, F a,b is a set of all monomials representable as a product of a fair monomials and b strongly fair monomials.
Remark 2.6.The notion of fair monomials was inspired from the conjectured construction of the initial ideal of ⟨x i , (x m ) (∞) ⟩ given in [Afsharijoo 2021, Conjecture 5.1].We use the notion to formulate concisely and prove the conjecture (see Theorem 3.3).
Example 2.7.The monomials of order at most two in F and F s are Using this, one can produce the monomials of order at most one in F 1,1 and F 2,0 Likewise, for the monomials of order at most two, we can write

Main results
The algebra of regular functions on the arc space of a fat point x m = 0 admits a natural filtration by subalgebras induced by the truncation of arcs.Our first main result, Theorem 3.1, gives a simple formula for the dimension of the subalgebra induced by the truncation at order h.Corollary 3.2 gives the generating series for these dimensions, as in (2).
Theorem 3.1.Let m and h be positive integers and k be a differential field of zero characteristic.Then Corollary 3.2.Let m be a positive integer and k be a differential field of zero characteristic.Then Given a polynomial ideal and monomial ordering, the monomials which do not appear as leading terms of the elements of the ideal are called standard monomials.Motivated by applications to combinatorics, Afsharijoo [2021, §5] used computations experiment to conjecture a description of the standard monomials of ⟨x m ⟩ (∞) with respect to the degree lexicographic ordering.Our second main result, Theorem 3.3, gives such a description and, combined with Lemma 4.10, establishes the conjecture.
Theorem 3.3.Let k be a differential field of zero characteristic.Consider a degree lexicographic monomial ordering on k[x (∞)  ] with the variables ordered as x < x ′ < x ′′ < • • • .Let m and i be positive integers with 1 ⩽ i ⩽ m.Then the set of standard monomials of the ideal ⟨x i , (x m ) (∞)  ⟩ is F i−1,m−i ; see Definition 2.5.Note that, for i = m, we obtain the differential ideal ⟨x m ⟩ (∞) .
Corollary 3.4.Theorem 3.3 also holds for the following orderings: • purely lexicographic with the variables ordered as in Theorem 3.3; • weighted lexicographic: monomials are first compared by the sum of the orders and then lexicographically as in Theorem 3.3.
Remark 3.5.The multiplicity of the scheme of polynomial arcs of degree less than h of x = 0, defined by ⟨x m , x (h) ⟩ (∞) , has been studied in [Ait El Manssour and Sattelberger 2023].It was shown that this multiplicity, equal to dim k k[x (∞)  ]/⟨x m , x (h) ⟩ (∞) , is a polynomial in m of degree h which is the Erhart polynomial of some lattice polytope [Ait El Manssour and Sattelberger 2023, Theorem 2.5].Theorem 3.1 together with a natural surjective morphism k[x (<h)  ]/⟨x m ⟩ ∞) implies that this polynomial is bounded by m h .Notation 4.1.Let k be a field.Then, for ξ = (ξ 0 , ξ 1 , . . ., ξ n ), we introduce a countable collection of symbols {ξ and by k (ξ (∞) ), we denote the exterior algebra of a k-vector space spanned by these symbols.k (ξ (∞) ) is equipped with a structure of a (noncommutative) differential algebra by The next proposition is a minor modification of [Pogudin 2014, Lemma 1].The proof we will give is a simplification of the proof in [Pogudin 2014, Lemma 1], which will be extended to a proof of Lemma 4.4.
Proposition 4.2.Let m be a positive integer.Consider η = (η 0 , . . ., η m−2 ) and ξ = (ξ 0 , . . ., ξ m−2 ).Let which is equipped with a structure of differential algebra (as a tensor product of differential algebras, using the Leibnitz rule, that is Then the kernel of ϕ is ⟨x m ⟩ (∞) .Example 4.3.Consider the case m = 3.Then we will have The image of x ′ will then be . This is a sum of tensor products of exterior polynomials of degree m in m − 1 variables, so it must be zero.Since (ϕ(x)) m = 0 and ϕ is a differential homomorphism, we conclude that Ker ϕ ⊃ ⟨x m ⟩ (∞) .Now we will prove the inverse inclusion.We define the weighted degree inverse lexicographic ordering ≺ on k[x (∞)  ] (see [Zobnin 2008, p. 524]): M ≺ N if and only if • tord M < tord N , where tord is defined as the sum of the orders, or • tord M = tord N and deg M < deg N , or • tord M = tord N , deg M = deg N , and N is lexicographically lower than M, where the variables are ordered For example, we will have Then, for every h ⩾ 0, the leading monomial of (x m ) (h) with respect to ≺ is (x (q) ) m−r (x (q+1) ) r , where q and r are the quotient and the reminder of the integer division of h by m, respectively.Let M be the set of all monomials not divisible by any monomial of the form (x (q) ) m−r (x (q+1) ) r .Then we can characterize M as We will define a linear map ψ from M to the set of monomials in with the following properties: (P1) For every P ∈ M, we have that ψ(P) ̸ = 0.
(P2) For every P ∈ M, the monomial ψ(P) appears in the polynomial ϕ(P) and, for any P 0 ∈ M such that P 0 ≺ P, the monomial ψ(P) does not appear in the polynomial ϕ(P 0 ).
Informally speaking, ψ(M) is the "leading monomial" in ϕ(M).Once such a map ψ has been defined, we can prove the proposition as follows: ∞) .By replacing Q with the result of the reduction of Q by x m , (x m ) ′ , . . .with respect to ≺, we can further assume that all the monomials in Q belong to M1 .Let Q 0 be the largest of them.By (P1) and (P2), ϕ(Q 0 ) will involve ψ(Q 0 ) and ϕ(Q − Q 0 ) will not, so ϕ(Q) ̸ = 0.This contradicts the assumption that Q ∈ Ker ϕ.The proposition is proved.
Therefore, it remains to define ψ satisfying (P1) and (P2).We will start with the case m = 2 to show the main idea while keeping the notation simple.We define ψ by where For proving (P1), we observe that, if h i+1 > h i + 1 for all i, then h so there are no coinciding ξ 's in (5).The construction for arbitrary m will consist of splitting the monomial into m − 1 interlacing submonomials and applying ( 5) with (η i , ξ i ) to i-th submonomial.More formally, if where a % b denotes the remainder of the division of a by b, and [α] denotes the integer part of α.Property (P1) is proved by applying (P1) for m = 2 to each submonomial.
We will prove that ϕ(P) involves ψ(P) by induction on deg P. The case deg P = 0 is clear.Consider P, with deg P > 0. Similarly to the preceding argument, one can obtain ψ(P) (from ψ(P/x (ℓ) )) only by taking η (i.e., the last term in ( 6)) from one of the occurrences of x (h ℓ ) in P. Therefore, the coefficient in front of ψ(P) in ϕ(P) will be, up to sign, equal to deg x (h ℓ ) times the coefficient in front of ψ(P/x (h ℓ ) ) in ϕ(P/x (h ℓ ) ).The latter is nonzero by the induction hypothesis.□ Lemma 4.4.In the notation of Proposition 4.2, let 1 ⩽ r < m.Then the preimage of the ideal in generated by Proof.We first prove that the image of x r belongs to ⟨η r −1 ⊗ ξ r −1 , . . ., η m−2 ⊗ ξ m−2 ⟩.This is because ϕ(x r ) is the sum of monomials which are products of r different η i ⊗ ξ i .Since there are m − 1 of them, every such monomial will involve at least one of the last m − r of the ⟩ and prove that ϕ(g) does not belong to ⟨η r −1 ⊗ ξ r −1 , . . ., η m−2 ⊗ ξ m−2 ⟩.We can assume that each monomial P of g belongs to We will use the map ψ defined in (6).In fact, ψ(P) does not involve the zero-order derivatives of ξ r −1 , . . ., ξ m−2 , since h i − [i/(m − 1)] can only be zero for a monomial in M only if i ⩽ r − 2. Thus, Assume that P 0 is the largest summand that appears in g.Then ϕ(P 0 ) involves ψ(P 0 ), but ϕ(g − P 0 ) does not.Therefore, ϕ(g) does not belong to ⟨η r −1 ⊗ ξ r −1 , . . ., η m−2 ⊗ ξ m−2 ⟩. □ 4.2.Upper bounds for the dimension.Throughout the section, we fix a differential field k of zero characteristic.
Proposition 4.5.Let m, h be positive integers.We denote by A m,h the subalgebra of k[x (∞)  ]/⟨x m ⟩ (∞) generated by the images of x, x ′ , . . ., x (h) .Then First we describe a general construction which will be a special case of the so-called associated graded algebra.Let A = A 0 ⊕ A 1 ⊕ A 2 ⊕ • • • be a ‫ޚ‬ ⩾0 -graded algebra over k equipped with a homogeneous derivation of weight one (that is, A ′ i ⊆ A i+1 for every i ⩾ 0).We introduce a map gr : A → A defined as follows: Consider a nonzero a ∈ A, and let i be the largest index such that a ∈ A i ⊕ A i+1 ⊕ • • • .Then we define gr(a) to be the image of the projection of a onto A i along A i+1 ⊕ A i+2 ⊕ • • • .In other words, we replace each element with its lowest homogeneous component.
Note that gr is not a homomorphism, it is not even a linear map.However, it has two important properties we state as a lemma.
Proof.To prove the first part, one sees that p does not vanish on the lowest homogeneous parts of a 1 , . . ., a n , so the homogeneity of the multiplication and derivation imply that taking the lowest homogeneous part commutes with applying p for a 1 , . . ., a n .
To prove the second part, let i be the lowest grading appearing among a 1 , . . ., a n .Restricting to the component of this weight, one gets a linear relation for gr(a 1 ), . . ., gr(a n ).□ Lemma 4.7.Let A be a graded differential algebra as above.Consider elements a 1 , . . ., a n in A, and denote the algebras (not differential) generated by a 1 , . . ., a n and gr(a 1 ), . . ., gr(a n ) by B and B gr , respectively.Then dim B gr ⩽ dim B.
Proof.The algebra B gr is spanned by all the monomials in gr(a 1 ), . . ., gr(a n ).We choose a basis in this spanning set, that is, we consider monomials p 1 , .
We define a grading on by setting the weights of η (i) j and ξ (i) j to be equal to i for every i ⩾ 0 and 0 ⩽ j < m − 1.The exterior algebra becomes a graded algebra, and the derivation is homogeneous of weight one.
We fix h ⩾ 0 and consider the following elements of : where ∂ is the operator of differentiation.We introduce and let Y h be the algebra generated by v 0 , . . ., v h .For every 0 ⩽ i ⩽ h, we have v m i = 0, so Y h is spanned by the products of the form Claim.There is an invertible (h + 1) × (h + 1) matrix M over ‫ޑ‬ such that, for u 0 , . . ., u h defined by we have (η j ⊗ ξ j ) (i) for every 0 ⩽ i ⩽ h.
We will first demonstrate how the proposition follows from the claim, and then we prove the claim.Since M is invertible, u 0 , . . ., u h generate Y h as well.Since gr(u 0 ), . . ., gr(u h ) generate A m,h , Lemma 4.7 implies that Therefore, it remains to prove the claim.For every 0 ⩽ i ⩽ h, we can write We set By expanding the binomial (1⊗1+(1⊗∂ +∂ ⊗1+∂ ⊗∂)) i , we can write v i = i j=0 i j u j .Then we have where M is the (h +1)×(h +1)-matrix with the (i, j)-th entry being i j .Since M is lower-triangular with ones on the diagonal, it is invertible.We set M := M −1 .So we have (u 0 , . . ., u h ) T := M(v 0 , . . ., v h ) T , which together with (8) finishes the proof of the claim. □ By combining the proof of Proposition 4.5 with Lemma 4.4, we can extend Proposition 4.5 as follows: Corollary 4.8.Let m, h, i be positive integers with 1 ⩽ i ⩽ m.By A (m,i),h we denote the subalgebra of k[x (∞)  ]/⟨x i , (x m ) (∞) ⟩ generated by the images of x, x ′ , . . ., x (h) .Then Proof.The proof will be a refinement of the proof of Proposition 4.5, and we will use the notation from there.Let π be the canonical homomorphism π : → ⟩ is homogeneous with respect to the grading on , there is a natural grading on i .
Proof.Suppose that P can be written as where each We first prove that we can make the product to be a product of nonoverlapping monomials.
Claim.For all 0 ⩽ i ⩽ m, we have h i,0 ⩽ r i,0 .
In the whole list of the h i, j , all the numbers to the right from h i,0 are ⩾ h i,0 .Therefore, after sorting, h i,0 will either stay or move to the left.Thus, h i,0 ⩽ r i,0 , so the claim is proved.
Hence if x (h i,0 ) • • • x (h i,ℓ i ) was a fair (respectively, strongly fair) monomial then x (r i,0 ) • • • x (r i,ℓ i ) is a fair (respectively, strongly fair) monomial.Now we will move all the strongly fair monomials to the right in the decomposition of P. We first prove that, for every where ) is a fair monomial, and Applying the described transformation while possible to the nonoverlapping decomposition of P, one can arrange that the last m − i components are strongly fair.□ Proposition 4.12.For every positive integers m, h, i with 0 ⩽ i ⩽ m, the cardinality of The proof of the proposition will use the following lemma: Lemma 4.13.For every integers h and d, we have ] and deg If one replaces F with F s , the cardinality will be h The map assigns to the orders of a monomial in ] a list of strictly increasing nonnegative integers not exceeding h.A direct computation shows that this map is a bijection.Since the number of such sequences of length d is equal to the number of subsets of [0, 1, . . ., h] of cardinality d, the number of monomials is h+1 d .The case of F s is analogous with the only difference being that the subset will be in [1, 2, . . ., h], thus yielding h d .□ Proof of Proposition 4.12.We will prove the proposition by induction on m.For the base case, we have F 0,0 = {1}, so the statement is true.Consider m > 0, and assume that for all smaller m the proposition is proved.We fix 0 ], let P 1 • • • P m be a decomposition from Lemma 4.10 with deg P m being as large as possible.We denote tail P := P m and head P := P 1 • • • P m−1 .
We will show that the map P → (head P, tail P) defines a bijection between F i,m−i and We will prove the case i < m, as the proof in the case i = m is analogous.First we will show that, for every P ∈ F i,m−i , we have ord head P ⩽ deg tail P. Assume the contrary, and let ℓ := ord head P > deg tail P.
This implies that x (ℓ) tail P ∈ F s .Thus, in the decomposition of Lemma 4.10, we could have taken P m to be x (ℓ) tail P.This contradicts the maximality of deg tail P. In the other direction, if We will now use the bijection (10) to count the elements in F i,m−i ∩ k[x (⩽h)  ].For i < m, For i = m: Thus, the proposition is proved.□ 4.4.Lower bounds for the dimension.
Notation 4.14.For a differential polynomial P ∈ k[x (∞)  ] and 1 ⩽ i ⩽ n, we define • tord x i P to be the total order of P in x i , that is, the largest sum of the orders of the derivatives of x i among the monomials of P; • deg x (∞) i P to be the total degree of P with respect to the variables x i , x ′ i , x ′′ i , . . . .
• We fix a monomial ordering ≺ on k[x (∞)  ] defined as follows: To each differential monomial and compare monomials by comparing the corresponding tuples lexicographically.
Definition 4.15 (isobaric ideal).An ideal )  ] is called isobaric if it can be generated by isobaric polynomials, that is, polynomials with all the monomials having the same total order.Proposition 4.16.For i = 1, 2, the elements of F i−1,2−i are the standard monomials modulo ⟨(x 2 ) (∞) , x i ⟩.Proof.We use Proposition 4.2 to obtain the differential homomorphism ϕ : k[x (∞)  ] → defined by ϕ(x) = η ⊗ ξ (we will use η and ξ instead of η 0 and ξ 0 for brevity).Let φ be the composition of ϕ with the projection onto /⟨η ⊗ ξ ⟩.We will prove the proposition for the elements in F 1,0 , and the other case can be done in the same way by replacing ϕ with φ.
, where h 0 ⩽ h 1 ⩽ • • • ⩽ h ℓ , be an element of F 1,0 .We will show that a summand appears in ϕ(X ) with nonzero coefficient.We will prove this by induction on ℓ.The base case ℓ = 0 is trivial, so let ℓ > 0. Since η (h 0 −ℓ) may come only from one of the occurrences of x (h 0 ) in X , we must take η (h 0 −ℓ) ⊗ ξ (ℓ) from one of the x (h 0 ) .Therefore, the coefficient at B(X ) in ϕ(X ) is deg x (h 0 ) X times the coefficient at B(X/x (h 0 ) ) in ϕ(X/x (h 0 ) ), which is nonzero by the induction hypothesis.
s ] be ideals, and we denote by M i the set of the standard monomials modulo I i with respect to degree lexicographic ordering for 1 ⩽ i ⩽ s.Then the standard monomials with respect to the ordering ≺ (see Notation 4.14) modulo ⟨I 1 , . . ., Proof.For each I i , consider the reduced Gröbner basis G i of I i with respect to the degree lexicographic ordering.For each pair f, g ∈ G := G 1 ∪ G 2 ∪ . . .∪ G s , their S-polynomial is reduced to zero by G • if f, g belong to the same G i , due to the fact that G i is a Gröbner basis; • otherwise, by the first Buchberger criterion (since f and g have coprime leading monomials).□ Proposition 4.18.
s ] be homogeneous and isobaric ideals (not necessarily differential).By M i we denote the set of standard monomials modulo I i with respect to the degree lexicographic ordering for 1 ⩽ i ⩽ s.We define a homomorphism (not necessarily differential) s and denote I := Ker(ϕ).Then the elements of are standard monomials modulo I with respect to the ordering ≺ (but maybe not all the standard monomials).
Proof.Consider a monomial P ∈ M, and fix a representation P = m 1 (x), . . ., m s (x) as in (12).Assume that P is a leading monomial of I .Then there exist monomials P 1 , . . ., P N such that P − N j=1 λ j P j ∈ Ker ϕ and ∀ 1 ⩽ j ⩽ N : P j ≺ P.
Claim.For every monomial m ̸ = m in ϕ(P), there exists 1 ⩽ j ⩽ s such that either deg y j (∞) m ̸ = deg y j (∞) m or tord y j m ̸ = tord y j m.
Assume the contrary, that there exists m such that, for every 1 ⩽ j ⩽ s, we have d i := deg y j (∞) m = deg y j (∞) m and tord y j m = tord y j m.We write m = m 1 (y 1 ) • • • m s (y s ).Let 1 ⩽ j ⩽ s be the largest index such that m j ̸ = m j .Since m j contains d j largest derivatives in m and has the same total order as m j , we conclude that m j = m j .Thus, the claim is proved.
We write the homogeneous and isobaric component of N j=1 λ j ϕ(P j ) of the same degree and total order in y i as m for every 1 ⩽ i ⩽ s as M i=1 µ i R i , where R i is a differential monomial and µ i ∈ k for every 1 ⩽ i ⩽ M. Then such a homogeneous and isobaric component of ϕ(P) − N j=1 λ j ϕ(P j ) is Q := m − M i=1 µ i R i due to the claim.Since, for every 1 ⩽ i ⩽ s, I s is homogeneous and isobaric, Q ∈ ⟨I 1 , . . ., I s ⟩.
Note that for every 1 ⩽ i ⩽ M, the differential monomial R i is a summand of ϕ(P j ) for some 1 ⩽ j ⩽ N .Thus, if P j = x (s 0 ) • • • x (s ℓ ) , then the derivatives that appear in the monomial R i are of orders s 0 , . . ., s ℓ .Hence, P j ≺ P implies R j ≺ m.Therefore, m is the leading monomial of Q contradicting Lemma 4.17.□ Corollary 4.19.The elements of F i−1,m−i are standard monomials modulo ⟨x i , (x m ) (∞)  ⟩.
□ Step 3: Return dim k[x (⩽h)  ]/(I (⩽H ) ∩ k[x (⩽h)  ]) .We expect the resulting bound to be exact (see also Question 5.1), for example, it is exact for I = ⟨x m ⟩.Our implementation of this algorithm in [Macaulay2] is available for download at the following webpage: https://mathrepo.mis.mpg.de/MultiplicityStructureOfArcSpaces.Table 2 shows some of the results we obtained.One can see that the computed dimensions form geometric series with the exponent being the multiplicity of the original ideal exactly as in Theorem 3.1.
However, we have also found ideals for which the generating series of the dimensions is definitely not equal to m/(1 − mt), where m is the multiplicity of the ideal.We show some examples of this type in Table 3.
Note that while Table 2 gives only indication that the generating series of the multiplicities for these ideals may be m/(1 − mt), Table 3 gives a proof that this is not the case for all the fat points.
Key technical tool: embedding into the exterior algebra.
. ., p N ∈ k[x 1 , . . ., x n ] such that p 1 gr(a 1 ), ..., gr(a n ) , ..., p N gr(a 1 ), ..., gr(a n ) form a basis of B gr .The first part of Lemma 4.6 implies that gr p i (a 1 , ..., a n ) = p i gr(a 1 ), ..., gr(a n )for every 1 ⩽ i ⩽ N .Then the second part of Lemma 4.6 implies that p 1 (a 1 , . .., a n ), . .., p N (a 1 , . .., a n ) are linearly independent.Since they belong to B, we have dim B ⩾ N = dim B gr .□Proof of Proposition 4.5.Let and ϕ be the exterior algebra and the homomorphism from Proposition 4.2.Proposition 4.2 implies that A m,h is isomorphic to the subalgebra of generated by

Table 2 .
(Bounds for) the dimensions of the truncations of the arc space.