Gr\"obner Bases for Increasing Sequences

Let $q,n \geq 1$ be integers, $[q]=\{1,\ldots, q\}$, and $\mathbb F$ be a field with $|\mathbb F|\geq q$. The set of increasing sequences $$ I(n,q)=\{(f_1,f_2, \dots, f_n) \in [q]^n:~ f_1\leq f_2\leq\cdots \leq f_n \} $$ can be mapped via an injective map $i: [q]\rightarrow \mathbb F $ into a subset $J(n,q)$ of the affine space ${\mathbb F}^n$. We describe reduced Gr\"obner bases, standard monomials and Hilbert function of the ideal of polynomials vanishing on $J(n,q)$. As applications we give an interpolation basis for $J(n,q)$, and lower bounds for the size of increasing Kakeya sets, increasing Nikodym sets, and for the size of affine hyperplane covers of $J(n,q)$.


Introduction
In the paper q, n ≥ 1 are integers, and [q] = {1, . . ., q}.We view [q] as an ordered set with 1 < 2 < • • • < q and consider the following set of increasing sequences Clearly we have as shown for example by a stars and bars argument.Let F be a field with |F| ≥ q.We can map [q] into a subset of F with the aid of an injective map i : [q] → F. This induces a map of I(n, q) to a subset J(n, q) of the affine space F n : a sequence (v 1 , . . ., v n ) ∈ [q] n is mapped to (i(v 1 ), . . ., i(v n )) ∈ F n .One may study the algebraic and geometric properties of the set J(n, q).We refer to [16], [18], [19] and references therein for research of this kind on other combinatorially relevant point sets and polynomial ideals.Most of the properties of J(n, q) considered here are independent of the map i, as long as it is an injection.One exception is grid maps.An injection i : [q] → F is a grid map if there exists an a ∈ F such that i(j) = a + j for j ∈ [q].In particular, a grid map of [q] exists iff the characteristic of F is at least q.Our main technical result is the determination of Gröbner bases, standard monomials and Hilbert functions (for the definitions please see Section 2) of the ideal I(J(n, q)) of polynomials vanishing on J(n, q) (Proposition 3.1, Corollary 3.2 and a useful interpolation basis for J(n, q)).We obtained applications of these results in several directions, as it is detailed next.

Interpolation and covering by hyperplanes
We denote by F[x 1 , . . ., x n ] = F[x] the polynomial ring over F with variables x 1 , . . ., x n .The standard monomials for the ideal I(J(n, q)) form a linear basis of the space of functions form J(n, q) to the ground field F.Here we exhibit an other natural basis of interpolation which turns out to be useful when we consider coverings by hyperplanes.The polynomials can be given quite explicitly when i is a grid embedding.
Theorem 1.1 Let s ∈ J(n, q).Then there exists a unique polynomial P s ∈ F[x] such that (i) P s (s) = 1 and P s (w) = 0 for each w ∈ J(n, q), w = s, with deg(P s ) = q − 1, and (ii) If i is a grid embedding, then we can write P s into the form P s = q−1 i=1 L i , where the L i are linear polynomials.
There are several results on covering discrete subsets of F n by hyperplanes (see [2], [4], [7], [6], and [22]), a prominent example being the theorem of Alon and Füredi [2] on the covers of discrete grids with the exception of a point.An analogous, sharp statement for increasing vectors is the following: Theorem 1.2 Let 0 < k ≤ n be an integer and let s 1 , . . ., s k ∈ J(n, q) be increasing vectors.Let {H j : 1 ≤ j ≤ m} be a set of affine hyperplanes such that J(n, q) \ {s 1 , . . ., A similar reasoning gives a sharp (non-constant) bound on the number of hyperplanes covering the whole J(n, q).Theorem 1.3 Let {H i : 1 ≤ i ≤ m} be a set of affine hyperplanes such that

Kakeya and Nikodym sets for increasing vectors
Let a, v ∈ F q n be vectors, with v = 0. Define the line ℓ(a, v) ⊆ F n q as ℓ(a, v) Wolff proposed the conjecture in [21], that for every ǫ > 0 and for every n ≥ 1 there exists a constant c(n, ǫ) such that for any Kakeya set K ⊆ F q n we have |K| ≥ c(n, ǫ)q n−ǫ .
Dvir obtained a stronger bound in his breakthrough work [10]: In [13] Ganesan obtained lower and upper bounds for the size of local Kakeya sets, where the set of required directions of the lines is local in the sense that it is possibly a proper subset of F q n \ {0}.We consider here some special local Kakeya sets, the so-called increasing Kakeya sets.
A subset K ⊆ F q n is an increasing Kakeya set, if for each 0 = v ∈ J(n, q) there exists an a ∈ F q n such that the line ℓ(a, v) ⊆ K. Clearly each Kakeya set is an increasing Kakeya set as well.We can prove the same lower bound as in Theorem 1.4 for increasing Kakeya sets.In fact, our results on the standard monomials and the Hilbert function of J(n, q) allow to use the argument of Dvir-Tao-Alon directly for increasing Kakeya sets.This is a strengthening of Dvir's result, as we have a smaller number of conditions (lines) to consider.Theorem 1.5 Let K ⊆ F q n be an increasing Kakeya set.Then It was proved on page 3 of [12], that for each prime power q there exists a subset K ⊆ F 2 q which is a union of q lines with different directions and with providing an optimal increasing Kakeya set in the case n = 2.We also have an optimal construction for n = q = 3, hence the bound of Theorem 1.5 is sharp if n = 2 or q = 2 or n = q = 3.
In general we have the following simple construction which seems to be good for small values of q: T (n, q) := ∪ 0 =v∈J(n,q) ℓ(0, v). ( Clearly we have It would be interesting to see, if there is a general upper bound on the size of the smallest increasing Kakeya sets, which is better than the best available upper bound for general Kakeya sets. Nikodym sets are closely related to Kakeya sets.A subset B ⊆ F q n is a Nikodym set, if for each z ∈ F q n there exists a line ℓ z ⊆ F q n through z such that ℓ z \ {z} ⊆ B. A variant of the Dvir-Alon-Tao-argument for Kakeya sets [10], (see also Theorem 2.9 in [15]) gives that the size of a Nikodym set B ⊆ F n q is at least q+n−2 n .
Here we obtain a version of this result for increasing vectors.We shall consider a local kind of Nikodym sets.B ⊆ F q n is an increasing Nikodym set if for each z ∈ J(n, q) there exists a line ℓ z through z such that ℓ z \ {z} ⊆ B.
Theorem 1.6 Let B ⊆ F q n be an increasing Nikodym set.Then The inequality of Theorem 1.6 strengthens the above bound for Nikodym sets in that here we have fewer conditions (lines to take care of).We note also that the set T (n, q) from ( 1) is an increasing Nikodym set.
In Section 2 we collected the preliminaries about Gröbner bases, standard monomials and Hilbert functions.In Section 3 we describe Gröbner bases and standard monomials for the vanishing ideal of increasing sequences J(n, q).We extend these results to some special subsets of J(n, q) and to strictly increasing sequences.Applications are discussed in Section 4.

Notation and results from Gröbner theory
A total ordering ≺ on the monomials x n is a term order, if 1 is the minimal element of ≺, and uw ≺ vw holds for any monomials u, v, w with u ≺ v. Important term orders are the lexicographic order ≺ l and the deglex order ≺ dl .We have The leading monomial lm(f ) of a nonzero polynomial f from the polynomial ring F[x] = F[x 1 , x 2 , . . ., x n ] is the largest monomial with respect to ≺, which has nonzero coefficient in the standard form of f .
Let I be an ideal of F[x].A finite subset G ⊆ I is a Gröbner basis of I if for every f ∈ I there exists a g ∈ G such that lm(g) divides lm(f ).It is known that such a G is a basis of I.A fundamental fact is (cf.[8, Chapter 1, Corollary 3.12] or [3, Corollary 1.6.5,Theorem 1.9.1]) that every nonzero ideal I of F[x] has a Gröbner basis with respect to any term order ≺.
A monomial w ∈ F[x] is a standard monomial for I if it is not a leading monomial of any f ∈ I. Let sm(I, ≺) stand for the set of all standard monomials of I with respect to the term-order ≺ over F. It is known (see [8, Chapter 1, Section 4]) that for a nonzero ideal I the set sm(I, ≺) is a basis of the F-vector space F[x]/I.In fact, every g ∈ F[x] can be written uniquely as g = h + f where f ∈ I and h is a unique F-linear combination of monomials from sm(I, ≺).For a subset X ⊆ F n we write I(X) for the ideal of polynomials from F[x] which vanish on X.When X ⊆ F n is a finite subset, interpolation gives that every X → F function is a polynomial function.The latter two facts imply for sm(X, ≺) A Gröbner basis {g 1 , . . ., g m } of I is reduced if the coefficient of lm(g i ) is 1, and no nonzero monomial in g i is divisible by any lm(g j ), j = i.By a theorem of Buchberger ([3, Theorem 1.8.7]) a nonzero ideal has a unique reduced Gröbner basis.
The initial ideal in(I) of I is the ideal in F[x] generated by the monomials {lm(f The notion of reduction is closely related to Gröbner bases.Let G be a Gröbner basis of and ideal I of F[x] and f ∈ F[x] be a polynomial.We can reduce f by the set G by subtracting multiples of polynomials g ∈ G from f in such a way that the resulting polynomial is composed of ≺-smaller monomials.It is known that this way any f ∈ F[x] can be reduced into a (unique) F-linear combination of standard monomials.This is related to the fact we have already mentioned: sm(I, ≺) provides a linear basis of F[x]/I.[5,Section 9.3]).It is easy to see that h F[x]/I (m) is the number of standard monomials of degree at most m, where the ordering ≺ is deglex. For is the dimension of the linear space of those X → F functions which are polynomials of degree at most m.

Gröbner bases for increasing sequences
Via an injective map i : [q] → F we consider [q] as a subset of our ground field F, in particular we assume that |F| ≥ q.We denote by J(n, q) the image of I(n, q) by the map induced by i: Let ≺ be a term order on the monomials of F[x] We say that an n-tuple (I 1 , . . ., I n ) of subsets of [q] is a good decomposition of [q], if (ii) if i < j, then x < y for each x ∈ I i , y ∈ I j .
In particular Next we define a polynomial f I 1 ,...,In (x) ∈ F[x] attached to a good decomposition (I 1 , . . ., I n ) of [q].We set
Proposition 3.1 G is the reduced Gröbner basis of J(n, q) for any term order ≺.Moreover for the standard monomials we have sm(J(n, q), ≺) = {x u : deg(x u ) ≤ q − 1}.
Proof.Observe first that the leading monomial of f I 1 ,...,In is x for any term order ≺ on the monomials of F[x], and it is of degree q.Moreover any monomial of degree q is the leading monomial of a polynomial of the shape f I 1 ,...,In for a suitable good decomposition (I 1 , . . ., I n ) of [q].It is then sufficient to prove that f I 1 ,...,In (x 1 , . . .x n ) vanishes on the vectors of J(n, q).Indeed then it follows that sm(J(n, q), ≺) ⊆ {x u : deg(x u ) ≤ q −1}.The two sets in the preceding formula have the same size n+q−1 q−1 , hence they must be equal: sm(J(n, q), ≺) = {x u : deg(x u ) ≤ q − 1}.
This gives the statement about the standard monomials and shows also that the polynomials in G indeed form a Gröbner basis.These polynomials are monic, and except for the leading monomial they are made of standard monomials.This proves that the Gröbner basis G is reduced.It remains to verify that if v ∈ J(n, q), then f I 1 ,...,In (v) = 0. Suppose that this is not the case, f I 1 ,...,In (v) = 0. Then straightforward induction on j gives that v j ∈ i(I 1 ∪ • • • ∪ I j ).This leads in the end to v n ∈ i([q]), a contradiction.
Please note that the Gröbner basis and the standard monomials are independent of the term order ≺ selected.For the the Hilbert function of J(n, q) we have: Proof.We apply Proposition 3.1 with ≺ being the deglex order.We obtain that the value of h J(n,q) (s) is the number of monomials in F[x] of degree at most s.
We obtain the following version of the Combinatorial Nullstellensatz [1] for increasing sequences.
Then there exists a vector v ∈ J(n, q) such that f (v) = 0.
Proof.By Proposition 3.1 f is a nontrivial linear combination of standard monomials, hence it can not vanish on the entire set J(n, q). .We now proceed to exhibit Gröbner bases for the ideals I(F ) where ∅ = F ⊆ J(n, q) is a downset in the sense that if have u, v ∈ J(n, q) with v ∈ F and u ≤ v (component-wise) then u ∈ F .We denote by F c the complement of F in J(n, q).We define first the map It is straightforward to see that φ is a bijection.
Please note that the Gröbner basis G is independent of the term order ≺.
Proof.We observe first that the polynomials in G all vanish on F .This was established for f I 1 ,...,In when (I 1 , . . ., I n ) ∈ T in Proposition 3.1.Consider now vectors v ∈ F and g ∈ I(n, q) such that i(g) ∈ F c .We have to establish f I 1 (g),...,In(g) (v) = 0.If f I 1 (g),...,In(g) (v) = 0, then an induction on j gives that v j ≥ i(g j ) holds for j = 1, . . ., n.This is immediate for j = 1, as v 1 can not be in i(I 1 (g)).Also for j > 1 the facts i(g j−1 ) ≤ v j−1 ≤ v j and v j ∈ i(I j (g)) give the claim.
On the other hand, by the selection of v and g there must be an index j such that i(g j ) > v j , giving a contradiction, which shows that the polynomials from G indeed vanish on F; here we used that F is a downset.
Next we verify that any monomial x u such that u ∈ φ(i −1 (F )) is divisible by the leading monomial of a polynomial from G. Suppose first that deg(x u ) ≥ q.Then there exists a good decomposition (I 1 , . . ., I n ) ∈ T and a w ∈ N n such that lm(x w • f (I 1 ,...,In) ) = x u .Suppose now that deg(x u ) < q.As u ∈ φ(i −1 (F )), we have u ∈ φ(i −1 (F c )).Let g := φ −1 (u) and consider f I 1 (g),...,In(g) whose leading monomial is x φ(g) = x u .We obtained We have equality here, as both sides have size |F|.Also, with G we can reduce any polynomial into a linear combination of standard monomials.This implies that G is a Gröbner basis.
We remark that the statement above implies also that if F ⊆ J(n, q) is a downset then φ(i −1 (F )) is a downset as well.This fact can be seen more directly, without using Gröbner theory, by the fact that φ −1 is an order preserving map.
Let q ≥ n ≥ 1 be fixed integers.Next we consider the collection of strictly increasing sequences.We put Clearly we have |SI(n, q)| = q n .Similarly to increasing sequences, we view [q] as an ordered subset of a field F via an injective map i : [q] → F, in particular we assume |F| ≥ q.We consider the image SJ(n, q) of SI(n, q) with respect to i: SJ(n, q) := {(i(g 1 ), . . ., i(g n )) : g ∈ SI(n, q)}.
The set SJ(n, q) is a subset of the affine space F n , and we proceed to determine a Gröbner basis for the ideal of polynomials from F[x] which vanish on SJ(n, q).Let ≺ denote a term order on the monomials of F[x].
We have a statement analogous to Proposition 3.1, with a similar proof.
Proof.We note first that for any monomial w ∈ F[x 1 , . . ., x n ] of degree q − n + 1 there exists a super decomposition (I 1 , . . ., I n ) of [q] such that the leading monomial of f I 1 ,...,In is w.We claim first that it suffices to verify that the polynomials f ∈ G all vanish on SJ(n, q).Indeed, then we obtain at once that sm(SJ(n, q), ≺) ⊆ {x u : deg(x u ) ≤ q − n}, which implies that the sets on the two sides are equal because they have the same size q n .We conclude from here as in Proposition 3.1, and obtain that G is a reduced Gröbner basis.
Applying the preceding result to a deglex order provides the Hilbert function of SJ(n, q).Corollary 3.6 h SJ(n,q) (s) = n + s s for each 0 ≤ s ≤ q − n.
Similarly to Corollary 3.3 we have a non-vanishing statement here as well.
Then there exists a v ∈ SJ(n, q) such that f (v) = 0.

Interpolation and covering
Proof of Theorem 1.1.From Proposition 3.1 we obtain that the function P s : J(n, q) → F can be obtained as a unique linear combination of standard monomials whose degree is at most q−1.The degree of this polynomial P s can not be smaller than q − 1, as otherwise the nonzero polynomial (x 1 − s 1 )P S would have degree ≤ q − 1 and vanish on J(n, q), which is impossible by Corollary 3.3.This proves (i).
We explain the proof for (ii) in the case when F = Q and i is the identical map of [q].The general case follows similarly, but involves more complicated notation.Let s = (s 1 , . . ., s n ) ∈ J(n, q) ⊆ [q] n .We shall give the linear factors L i of P s in terms of s.We include the linear polynomials x 1 − t for each t ∈ [q] such that t < s 1 and x n − t for each t > s n .There are as many as q − (s n − s 1 + 1) such linear factors.Moreover, for each i > 1 such that s i − s i−1 = k > 0, we consider the k polynomials There are s n − s 1 linear polynomials of this type.The product Q of the preceding q−1 linear polynomials does not vanish on s.Suppose that t ∈ J(n, q) is a vector such that Q(t) = 0. Then an inductive argument proceeding from i = 1 to i = n shows that t ≥ s (component-wise comparison).A similar argument in the direction from i = n to i = 1 gives t ≤ s, hence t = s.A suitable scalar multiple of Q will be appropriate as P s .Uniqueness follows as in the general case.
Example.Let n = 5, q = 5, and s = (1, 2, 2, 4, 4).Then we have Remarks. 1.The polynomials Q and P s in (ii) are explicitly determined by s in the proof.2. The uniqueness of P s in (i) can also be established by a dimension counting argument.The polynomials P s form a basis the space of J(n, q) → F functions and hence also of the space of polynomials of degree at most q − 1 in F[x].In particular no nonzero polynomial from the latter space can be the identically 0 function on J(n, q).This reasoning gives also an alternative proof of the part of Proposition 3.1 on standard monomials, when ≺ is the deglex order, and then of Corollary 3.2.Lemma 4.1 Let 0 < k ≤ n be an integer and let s 1 , . . ., s k ∈ J(n, q) be increasing vectors.Let P ∈ F[x] be a polynomial such that P (s i ) = 0 for each i and P (w) = 0 whenever w ∈ J(n, q) \ {s 1 , . . ., s k }.Then deg(P ) ≥ q − 1.
Proof.Suppose for contradiction that there exists a polynomial P ∈ F[x] such that P (s i ) = 0 for each i, but P (w) = 0 for each w ∈ J(n, q) \ {s 1 , . . ., s k }, and deg(P ) < q − 1.There exists a hyperplane in F n which contains the points s i , that is, a linear polynomial L ∈ F[x] such that L(s i ) = 0 for each i.We define the nonzero polynomial Q := P • L. Then Q(w) = 0 for each w ∈ J(n, q).But deg(Q) ≤ q − 1, which contradicts to Corollary 3.3.
Proof of Theorem 1.2.Let L j ∈ F[x] be a linear polynomial whose set of zeros is exactly H j .We can apply Lemma 4.
Proof of Theorem 1.3.Let L i be a linear polynomial defining H i .Then P = L 1 • • • L m vanishes on J(n, q) and deg P = m.Here m < q would contradict to Corollary 3.3.
From this we obtain that P D (v) = 0 for each v ∈ T .This contradicts to (3), as P D is a nontrivial linear combination of standard monomials, hence can not vanish on the entire T .This finishes the proof.
Proof of Theorem 1.6.Suppose for contradiction that there exists an increasing Nikodym set B with size Then there exists a nonzero polynomial P ∈ F q [x] such that deg(P ) ≤ q − 2 and P (v) = 0 for each v ∈ B. Let z ∈ J(n, q) ⊂ F n q be an arbitrary element.Then there exists a line ℓ z = {z + tv : t ∈ F q } with 0 = v ∈ F n q through z such that ℓ z \ {z} ⊆ B. Define the polynomial Q(t) := P (z + tv) ∈ F q [t].Then Q(t) = 0 for each t ∈ (F q ) * , because ℓ z \ {z} ⊆ B. It follows from deg(Q) ≤ q − 2 that Q is the identically 0 polynomial, hence P (z) = Q(0) = 0. We obtained that P (z) = 0 for each z ∈ J(n, q), and then deg(P ) ≤ q − 2 implies that P is the identically 0 polynomial by Proposition 3.1.This contradiction proves the claim.