Algebraic stories from one and from the other pockets

We present a number of questions in commutative algebra posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014 - Spring 2017.


The Waring problem for complex-valued forms
The following famous result on binary forms was proven by J. J. Sylvester in 1851. (Below we use the terms "forms" and "homogeneous polynomials" as synonyms.) Theorem 1.1 (Sylvester's Theorem [Sy51]). (i) A general binary form f of odd degree k = 2s − 1 with complex coefficients can be written as (ii) A general binary form f of even degree k = 2s with complex coefficients can be written as f (x, y) = λx k + s j=1 (α j x + β j y) k .
Sylvester's result was the starting point of the study of the so-called Waring problem for polynomials which we discuss below.
Let S = C[x 1 , . . . , x n ] be the polynomial ring in n variables with complex coefficients. With respect to the standard grading, we have S = d≥0 S d , where S d denotes the vector space of all forms of degree d.
Definition 1.2. Let f be a form of degree k in S. A presentation of f as a sum of k-th powers of linear forms, i.e., f = l k 1 + . . . + l k s , where l 1 , . . . , l s ∈ S 1 , is called a Waring decomposition of f . The minimal length of such a decomposition is called the Waring rank of f , and we denote it as rk(f ). By rk • (k, n) we denote the Waring rank of a general complex-valued form of degree k in n variables.
Remark 1.3. Besides being a natural question from the point of view of algebraic geometry, the Waring problem for polynomials is partly motivated by its celebrated prototype, i.e., the Waring problem for natural numbers. The latter was posed in 1770 by a British number theorist E. Waring who claimed that, for any positive integer k, there exists a minimal number g(k) such that every natural number can be written as sum of at most g(k) k-th powers of positive integers. The famous Lagrange's four-squares Theorem (1770) claims that g(2) = 4 while the existence of g(k), for any The title alludes to the famous collection of mystery novels "Povídky z jedné a z druhé kapsy" by K.Čapek.
integer k ≥ 2, is due to D. Hilbert (1900). Exact values of g(k) are currently known only in a few cases.
1.1. Generic k-rank. In terms of Definition 1.2, Sylvester's Theorem claims that the Waring rank of a general binary complex-valued form of degree k equals k 2 . More generally, the important result of J. Alexander and A. Hirschowitz [AH95] completely describes the Waring rank rk • (k, n) of general forms of any degree and in any number of variables.
Going further, R.F. and B.S. jointly with G. Ottaviani considered the following natural version of the Waring problem for complex-valued forms, see [FOS12].
Definition 1.5. Let k, d be positive integers. Given a form f of degree kd, a k-Waring decomposition is a presentation of f as a sum of k-th powers of forms of degree d, i.e., f = g k 1 + · · · + g k s , with g i ∈ S d . The minimal length of such an expression is called the k-rank of f and is denoted by rk k (f ). We denote by rk • k (kd, n) the k-rank of a general complex-valued form of degree kd in n variables.
In this notation, the case d = 1 corresponds to the classical Waring rank, i.e., if k = deg(f ), then rk(f ) = rk k (f ) and rk • (k, n) = rk • k (k, n). Since the case k = 1 is trivial, we assume below that k ≥ 2.
Problem A. Given a triple of positive integers (k, d, n), calculate rk • k (kd, n). The main result of [FOS12] states that, for any triple (k, d, n) as above, At the same time, by a simple parameter count, one has an obvious lower bound for rk • (k, n) given by A remarkable fact about the upper bound given by (2) is that it is independent of d. Therefore, since the right-hand side of (3) equals k n when d ≫ 0, we get that for large values of d, the bound in (2) is actually sharp. As a consequence of this remark, for any fixed n ≥ 1 and k ≥ 2, there exists a positive integer d k,n such that rk • k (kd, n) = k n , for all d ≥ d k,n . In the case of binary forms, it has been proven that (3) is actually an equality [Re13,LORS17]. Exact values of d k,n , and the behaviour of rk k (kd, n) for d ≤ d k,n , have also been computed in a few other cases, see [On16, Section 3.3]. These results agree with the following illuminating conjecture suggested by G. Ottaviani in 2014.
Conjecture 1.6. The k-rank of a general form of degree kd in n variables is given by Observe that, for k ≥ 3, Conjecture 1.6 claims that the naïve bound (3) obtained by a parameter count is actually sharp, while, for k = 2, due to an additional group action there are many defective cases where the inequality is strict.
Remark 1.7. Problems about additive decompositions including the above Waring problems can be usually rephrased geometrically in terms of secant varieties. In the case of k-Waring decompositions, we need to consider the variety of powers V n,kd . In other words, it is the closure of the set of forms whose k-rank is at most s. Since the variety of powers is non-degenerate, i.e., it is not contained in any proper linear subspace of the space of forms of degree kd, the sequence of secant varieties stratifies the latter space and coincides with it for all sufficiently large s. Hence, the k-rank of a general form is the smallest value of s for which the s-th secant variety of V  In [Re13, Theorem 5.4], B. Reznick proved that the maximal Waring rank of binary forms of degree k equals k. Moreover, the maximal value k is attained exactly on the binary forms representable as ℓ 1 ℓ k−1 2 , where ℓ 1 and ℓ 2 are any two non-proportional linear binary forms. (Apparently these claims have been known much earlier, but have never been carefully written down with a complete proof.) Problem B. Given a triple of positive integers (k, d, n), calculate rk max k (kd, n).
At the moment, we have an explicit conjecture about the maximal k-rank only in the case of binary forms. Conjecture 1.9. For any positive integers k, d, the maximal k-rank rk max k (kd, 2) of binary forms equals k. Additionally, in the above notation, binary forms representable by ℓ 1 ℓ kd−1 2 , where ℓ 1 and ℓ 2 are non-proportional linear forms, have the latter maximal k-rank. Conjecture 1.9 is obvious for k = 2 since, for any binary form f of degree 2d, we can write The first non-trivial case is the one of binary sextics, i.e., k = 3, d = 2, which has been settled in [LORS17] where it has also been shown that the 4-rank of x 1 x 7 2 is equal to 4. Remark 1.10. The best known general result about maximal ranks is due to G. Bleckherman and Z. Teitler, see [BT15] where they prove that the maximal rank is always at most twice as big as the generic rank. (This fact is true both for the classical (d = 1) and for the higher (d ≥ 2) Waring ranks.) In the classical case of Waring ranks, this bound is (almost) sharp for binary forms, but in many other cases it is rather crude. At present, better bounds are known only in few special cases of low degrees [BD13,Je14]. To the best of our knowledge, the exact values of the maximal Waring rank are only known for binary forms (classical, see [Re13]), quadrics (classical), ternary cubics (see [Seg42,LT10]), ternary quartics [Kl99], ternary quintics [DeP15] and quaternary cubics [Seg42].
1.3. The k-rank of monomials. Let m = x a1 1 · · · x an n be a monomial with 0 < a 1 ≤ a 2 ≤ · · · ≤ a n . It has been shown in [CCG12] that the classical Waring rank of m is equal to 1 (a1+1) i=1,...,n (a i +1). Later E. Carlini and A.O. settled the case of the 2-rank, see [CO15]. Namely, if m is a monomial of degree 2d, then we can write m = m 1 m 2 , where m 1 and m 2 are monomials of degree d. From identity (5), it follows that the 2-rank of m is at most two. On the other hand, m has rank one exactly when we can choose m 1 = m 2 , i.e., when the power of each variable in m is even.
While the cases k = 1 and k = 2 are solved, for k ≥ 3, the question about the k-rank of monomials of degree kd, is still open. At present, we are only aware of two general results in this direction. Namely, [CO15] contains the bound rk k (m) ≤ 2 k−1 , and recently, S.L., A.O., B.S., together with B. Reznick, have shown that rk k (m) ≤ k when d ≥ n(k − 2), see [LORS17]. Thus, for fixed k and n, all but a finite number of monomials of degree divisible by k have k-rank less than k.
Problem C. Given k ≥ 3 and a monomial m of degree kd, determine the monomial k-rank rk k (m).
In the case of binary forms, a bit more is currently known which motivates the following question.
Problem D. Given k ≥ 3 and a monomial x a y b of degree a + b = kd, it is known that rk k (x a y b ) ≤ max(s, t) + 1, where s and t are the remainders of the division of a and b by k, see [CO15]. Is it true that the latter inequality is, in fact, an equality?
1.4. Degree of the Waring map. Here again, we concentrate on the case of binary forms (i.e., n = 2). As we mentioned above, in this case, it is proven that Definition 1.11. We say that a pair (k, d) is perfect if kd+1 d+1 is an integer. All perfect pairs are easy to describe.
In each E j , the corresponding quotient equals jd + 1.
Let W k,d be the same map, but defined up to a permutation of the g i 's. We call it the Waring map. By [LORS17, Theorem 2.3], W k,d is a generically finite map of complex linear spaces of the same dimension. By definition, its degree is the cardinality of the inverse image of a generic form in S kd .
Problem E. Calculate the degree of W k,d for perfect pairs (k, d).
For the classical Waring decomposition (d = 1), we have a perfect pair if and only if k is odd. From Sylvester's Theorem, we know that in this case the degree of the Waring map is 1, i.e., the general binary form of odd degree has a unique Waring decomposition, up to a permutation of its summands.
Remark 1.13. For the case of the classical Waring decomposition, the latter problem has also been considered in the case of more variables. In modern terminology, the cases where the general form of a given degree has a unique decomposition up to a permutation of the summands are called identifiable. Besides the case of binary forms of odd degree, some other identifiable cases are classically known. These are the quaternary cubics (Sylvester's Pentahedral Theorem [Sy51]) and the ternary quintics [Hilb,Pal03,Ri04,MM13]. Recently, F. Galuppi and M. Mella proved that these are the only possible identifiable cases, [GM16].
Remark 1.14. Problems dealing with additive decompositions of homogeneous polynomials similar to those we consider in this section, have a very long story going back to J. J. Sylvester and the Italian school of algebraic geometry of the late 19-th century. In the last decades, these problems received renewed attention due to their potential applications. Namely, homogeneous polynomials can be naturally identified with symmetric tensors and in several applied branches of science where such tensors are used, for example, to encode multidimensional data sets, additive decompositions of tensors play a crucial role as an efficient way to code those. We refer to [La12] for an extensive exposition of these connections.

Ideals of generic forms
Let I be a homogeneous ideal in S, i.e., an ideal generated by homogeneous polynomials. The ideal I and the quotient algebra R = S/I inherit the grading of the polynomial ring.
Definition 2.1. Given a homogeneous ideal I ⊂ S, we call the function is called the Hilbert series of R.
Let I be a homogeneous ideal generated by forms f 1 , . . . , f r of degrees d 1 , . . . , d r , respectively. It was shown in [FL90] that, for fixed parameters (n, d 1 , . . . , d r ), there exists only a finite number of possible Hilbert series for S/I, and that there is a Zariski open subset in the space of coefficients of the f i 's on which the Hilbert series of S/I is one and the same and, in the appropriate sense, it is minimal among all possible Hilbert series, see below. We call algebras with this Hilbert series generic. There is a longstanding conjecture about this minimal Hilbert series formulated by the first author, see [Fr85].
Conjecture 2.2 (Fröberg's Conjecture, 1985). Let f 1 , . . . , f r be generic forms of degrees d 1 , . . . , d r , respectively. Then the Hilbert series of the quotient algebra R = S/(f 1 , . . . , f r ) is given by (6) In other words, [ i≥0 a i z i ] + is the truncation of a power series at its first non-positive coefficient.
Conjecture 2.2 has been proven in the following cases: for r ≤ n (easy exercise, since in this case I is a complete intersection); for n ≤ 2, [Fr85]; for n = 3, [An86], for r = n + 1, which follows from [St78]. Additionally, in [HL87] it has been proven that (6) is correct in the first nontrivial degree min r i=1 (d i + 1). There are also other special results in the case d 1 = · · · = d r , see [Ni17,Au95,FH94,MM03,Ne17]. We should also mention that [FL17] contains a survey of the existing results on the generic series for various algebras and also it studies the (opposite) problem of finding the maximal Hilbert series for fixed parameters (n, d 1 , . . . , d r ).
It is known that the actual Hilbert series of the quotient ring of any ideal with the same numerical parameters is lexicographically larger than or equal to the conjectured one. This fact implies that if for a given discrete data (n, d 1 , . . . , d r ), one finds just a single example of an algebra with the Hilbert series as in (6), then Conjecture 2.2 is settled in this case.
Although algebras with the minimal Hilbert series constitute a Zariski open set, they are hard to find constructively. We are only aware of two explicit constructions giving the minimal series in the special case r = n+ 1, namely R. Stanley's choice x d1 1 , . . . , x dn n , (x 1 + · · · + x n ) dn+1 , and C. Gottlieb's choice x d1 1 , . . . , x dn n , h dn+1 , where h d denotes the complete homogeneous symmetric polynomial of degree d, (private communication). To the best of our knowledge, already in the next case r = n + 2 there is no concrete guess about how to construct a similar example. There is however a substantial computer-based evidence pointing towards the possibility of replacing generic forms of degree d by a product of generic forms of much smaller degrees. We present some problems and conjectures related to such pseudo-concrete constructions below.
2.1. Hilbert series of generic power ideals. Differently from the situation occurring in R. Stanley's result, if we consider ideals generated by more than n + 1 powers of generic linear forms, there are known examples of (n, d 1 , ..., d r ) for which algebras generated by powers of generic linear forms fail to have the Hilbert series as in (6).
Recall that ideals generated by powers of linear forms are usually called power ideals. Due to their appearance in several areas of algebraic geometry, commutative algebra and combinatorics, they have been studied more thoroughly. In the next section, we will discuss their relation with the so-called fat points. (For a more extensive survey of power ideals, we refer to a nice paper by F. Ardila and A. Postnikov [AP10].) Studying Hilbert functions of generic power ideals, A. Iarrobino formulated the following conjecture, usually referred to as the Fröberg-Iarrobino Conjecture, see [Ia97,Ch05]. This conjecture is still largely open. In [FH94] R.F. and J. Hollman checked it for low degrees and low number of variables using the first version of the software package Macaulay2. In the last decades, some progress has been made in reformulation of Conjecture 2.3 in terms of the ideals of fat points and linear systems. We will return to this topic in the next section.

Hilbert series of other classes of ideals.
Computer experiments suggest that in order to always generically get the Hilbert function as in (6) we need to replace power ideals by slightly less special ideals.
For example, given a partition µ = (µ 1 , . . . , µ k ) ⊢ d, we call by a µ-power ideal an ideal generated by forms of the type (l µ 1 , . . . , l µ r ), where l µ i = l µ1 i,1 · · · l µ k i,k and l i,j 's are distinct linear forms. Problem F. For µ = (d), does a generic µ-power ideal have the same Hilbert function as in (6)?
Performed computer experiments suggest a positive answer to the latter problem. L. Nicklasson has also conjectured that ideals generated by powers of generic forms of degree ≥ 2 have the Hilbert series as in (6).

Conjecture 2.4 ([Ni17]
). For generic forms g 1 , . . . , g r of degree d > 1, the ideal (g k 1 , . . . , g k r ) has the same Hilbert series as the one generated by r generic forms of degree dk.
It was observed in [LORS17, Theorem A.3] that Conjecture 2.4 implies Conjecture 1.6, connecting the two first sections of the present paper. It was also shown that Conjecture 2.4 holds in the case of binary form by specializing the g i 's to be d-th powers of linear forms and applying the fact that generic power ideals in two variables have the generic Hilbert series [GS98]. The same idea gives a positive answer to Problem F in the case of binary forms, by specializing l i,1 = . . . = l i,k , for i = 1, . . . , r.
2.3. Lefschetz properties of graded algebras. We say that a graded algebra A has the weak Lefschetz property (WLP) (respectively, the strong Lefschetz property (SLP)) if the multiplication map ×l : A i → A i+1 (respectively, ×l k : A i → A i+k ) has the maximal rank, i.e., it is either injective or surjective, for a generic linear form l and all i (resp., for all i and k). (For more references and open problems about the Lefschetz properties, see [MN03].) Problem G. It has been conjectured that each complete intersection R = S/(f 1 , . . . , f n ) satisfies the WLP and also the SLP, see [HMNW03]. Does the same hold for R = S/(f 1 , . . . , f r ), with f 1 , . . . , f r being generic forms, and r > n?
It follows from [St80] that monomial complete intersections satisfy the SLP. In [BFL18] the following situation has been studied. For the ring T n,d,k = S/(x d 1 , . . . , x d n ) k , it is shown that for k ≥ d n−2 , n ≥ 3, (n, d) = (3, 2), T n,d,k fails the WLP. For n = 3, there is an explicit conjecture when the WLP holds. Additionally, there is some information about n > 3.
Problem H. When are the WLP and the SLP true for T n,d,k ?
We now introduce the concept of the µ-Lefschetz properties. Let µ = (µ 1 , . . . , µ k ) be a partition of d, i.e., k i=1 µ i = d. We say that an algebra has the µ-Lefschetz property if ×l µ : A i → A i+d has maximal rank for all i, where l µ = l µ1 1 · · · l µ k k , and l i 's are generic linear forms. Problem I. For R = S/(f 1 , . . . , f r ), where f 1 , . . . , f r are generic forms, does R satisfy the µ-Lefschetz property for all partitions µ?

Symbolic powers
For a prime ideal ℘ in a Noetherian ring R, define its m-th symbolic power ℘ (m) as It is the ℘-primary component of ℘ m . For a general ideal I in R, its m-th symbolic power is defined as I (m) = ∩ ℘∈Ass(I) (I m R ℘ ∩ R).

Hilbert functions of fat points.
Let I X be the ideal in C[x 1 , . . . , x n ] defining a scheme of reduced points X = P 1 + . . . + P s in P n−1 , say I X = ℘ 1 ∩ . . . ∩ ℘ s where ℘ i is the prime ideal defining the point P i . Then, the m-th symbolic power I (m) is the ideal I (m) X = ℘ m 1 ∩ . . . ∩ ℘ m s which defines the scheme of fat points X = mP 1 + . . . + mP s .
Ideals of 0-dimensional schemes are classical objects of study since the beginning of the last century. Their Hilbert functions are of particular interest. Study of these ideals and calculation of their Hilbert functions can be often related to the so-called polynomial interpolation problem. Indeed, the homogeneous part of degree d of the ideal I X is the space of hypersurfaces of degree d in P n−1 passing through the P i 's up to order m − 1, i.e., the space of polynomials of degree d whose partial differentials up to order m − 1 vanish at every P i .
It is well-known that the Hilbert function of 0-dimensional schemes is strictly increasing until it reaches the multiplicity of the scheme, see [IK06, Theorem 1.69]. Hence, since the degree of a m-fat point in P n−1 is n−1+m−1 n−1 , the expected Hilbert function is In the case of simple generic points, i.e., for m = 1, it is known that the actual Hilbert function is as expected.
In the case of double points (m = 2), counterexamples were known since the end of the 19-th century. In 1995, after a series of important papers, J. Alexander and A. Hirschowitz proved that the classically known examples were the only counterexamples. For higher multiplicity, very little is known at present. In the case of projective plane, a series of equivalent conjectures have been given by B. Segre [Seg61], B. Harbourne [Har86], A. Gimigliano [Gim87] and A. Hirschowitz [Hir89]. These are known as the SHGH-Conjecture, see [Har00] for a survey of this topic.
Apolarity Theory is a very useful tool in studying of the ideals of fat points and it is connecting all the algebraic stories we have told above. In particular, the following lemma is crucial. (We refer to [IK06] and [Ger96] for an extensive description of this issue.) Lemma 3.1 (Apolarity Lemma). Let X = P 1 + . . . + P s be a scheme of reduced points in P n−1 and let L 1 , . . . , L s be linear forms in C[x 0 , . . . , x n ] such that, for any i, the coordinates of P i are the coefficients of L i . Then, for every m ≥ d, Using this statement we obtain that calculation of the Hilbert function of a scheme of fat points is equivalent to the calculation of the Hilbert function of the corresponding power ideal. In particular, Fröberg-Iarrobino conjecture (Conjecture 2.3) can be rephrased as a conjecture about the Hilbert function of ideals of generic fat points.
Recently R.F. raised the question about what happens in case of the ideals of generic fat points in a multi-graded space. A point in multi-projective space P ∈ P n1−1 ×. . .×P nt−1 is defined by a prime ideal ℘ in the multi-graded polynomial ring S = C[x 1,1 , . . . , x 1,n1 ; . . . ; x t,1 , . . . , x t,nt ] = I⊂N t S I , where S I is the vector space of multi-graded polynomials of multi-degree I = (i 1 , . . . , i t ) ∈ N t . A scheme of fat points X = mP 1 + . . . + mP s is the scheme associated with the multi-graded ideal ℘ m 1 ∩ . . . ∩ ℘ m s . Problem J. Given a scheme of generic fat points X ⊂ P n1−1 ×. . .×P nt−1 , what is the multi-graded Hilbert function HF S/IX (I), for I ∈ N t ?
This question was first considered by M. V. Catalisano, A. V. Geramita and A. Gimigliano who solved it in the case of double points, i.e., for m = 2 in P 1 × . . . × P 1 . Recently, A.O. jointly with E. Carlini and M. V. Catalisano resolved the case of triple points (m = 3) in P 1 × P 1 and computed the Hilbert function for an arbitrary multiplicity except for a finite region in the space of multi-indices, see [CCO17].
3.2. Symbolic powers vs. ordinary powers. As we mentioned above, if I is the ideal defining a set X of points, the m-th symbolic power of I is the ideal of polynomials vanishing up to order m − 1 at all points in X or, in other words, the space of hypersurfaces which are singular at all points in X up to order m − 1. For this reason, symbolic powers are interesting from a geometrical point of view, but they are more difficult to study compared to the usual powers which carry less geometrical information. Hence, it is important to find relations between them. Observe that the inclusion I m ⊂ I (m) is trivial.
Containment problems between the ordinary and the symbolic powers of ideals of points have been studied in substantial details. One particularly interesting question is to understand for which pairs of positive integers (m, r), I (m) ⊂ I r . A very important result in this direction is the fact that, for any ideal I of reduced points in P n and any r > 1, we have I (nr) ⊂ I r . This statement was proven in [ELS01] by L. Ein, R. Lazersfeld and K. Smith for characteristic 0 and by M. Hochster and C. Huneke in positive characteristic, see [HH02]. At present, the important question is whether the bound in the latter statement is sharp. In [DSTG13], M. Dumicki, T. Szemberg and H. Tutaj-Gasińska provided the first example of a configuration of points such that I (3) ⊂ I 2 . (We refer to [SS17] for a complete account on this topic.) In the recent paper [GGSVT16], F. Galetto

4.2.
Non-negative forms. The next circle of problems is related to the celebrated article [Hil88] of D. Hilbert and to a number of results formulated in [CLR80].
Denote by P n,m the set of all non-negative real forms, i.e., real homogeneous polynomials of (an even) degree m in n variables which never attain negative values; denote by Σ n.m ⊆ P n,m the subset of non-negative forms which can be represented as sums of squares of real forms of degree n 2 . (In [Hil88] D. Hilbert proved that ∆ n,m = P n,m \ Σ n,m is non-empty unless the pair (n, m) is of the form (n, 2), (2, m) or (4, 3).) Finally, if Z(p) stands for the real zero locus of a real form p, denote by B n,m (resp. B ′ n,m ) the supremum of |Z(p)| over p ∈ P n,m such that |Z(p)| < ∞ (resp. over p ∈ Σ n,m such that |Z(p)| < ∞). In other words, B(n, m) is the supremum of the number of zeros of non-degenerate forms under the assumption that all these roots are isolated (and similarly for B ′ n,m ). Obviously, B ′ n,m < B n,m . The following basic question was posed in [CLR80].
Problem L. Are B n,m and B ′ n,m finite for any pair (n, m) with even n? In [CLR80] it was shown that the answer to this problem is positive for m = 2, 3 and for the pair (4, 4). Relatively recently, in [CS13] the following upper bound for B n,m was established However this bound can not be sharp, as shown in [Ko17]. In case of B ′ n,m , the following guess seems quite plausible and is proven for m = 3. For B n,m , no similar guess is known, but some intriguing information is available in the case m = 3, see [CLR80]. The following problem is related to the classical Petrovski-Oleinik upper bound on the number of real ovals of real plane algebraic curves.
Problem M. Determine lim n→∞ Bn,3 n 2 . The latter limit exists and lies in the interval 5 18 , 1 2 , see [CLR80]. 4.3. Polynomial generation. Let p be a prime number and let F p denote the field with p elements. Consider the two maps Here x a := x a1 1 · · · x an n , where each a i is regarded as an integer, and Z(f ) is the zero locus of f in F n p , i.e., Z(f ) := {a ∈ F n p | f (a) = 0}. When p = 2, then φ is a bijective map on the vector space of polynomials of degree at most one in each variable, and φ 4 (f ) = f , see [Lu15]. The map ψ, suggested by M. Boij, is a linear bijective map on the vector space of polynomials with degree at most p − 1 in each variable, and when p = 2, these two maps are closely related in the sense that φ(f ) = ψ(f ) + a∈F n 2 x a . Consider now the case n = 1 and p > 2. The map φ is no longer a bijection, but the sequence φ(f ), φ 2 (f ), . . . will eventually become periodic. It is an easy exercise to show that 0 → 1 + x + · · · + x p−1 → x → 1 → 0. When p ≤ 17, this is the only period, i.e., φ(f ) d(f ) = 0 for some d(f ). For p = 71, we have found a period of length two; 1+x 63 → x 23 +x 26 +x 34 +x 39 +x 41 +x 51 +x 70 → 1+x 63 . One can show that the length of the period is always an even number, but it is not clear which even numbers that can occur as lengths of periods.
Problem N. For n = 1 and given p, what are the (lengths of the) possible periods of φ?
Let us now turn to the map ψ and the case n = 1. For p = 3, ψ 8 (f ) = f for all polynomials f in F 3 [x] of degree at most two. For p = 5, the least i such that ψ i = Id on the space of polynomials of degree at most four, is equal to 124. For p = 7, the corresponding number is 1368.
Problem O. For n = 1 and given p, find the minimal positive integer i such that ψ i is the identity map on the space of polynomials of degree at most p − 1.

4.4.
Exterior algebras. Let f be a generic form of even degree in the exterior algebra E over C with n generators. Moreno and Snellman showed that the Hilbert series of E/(f ) is equal to the expected series [(1 + t) d (1 − t d )] + , see [MS02]. When the degree of f is odd, we have (f ) ⊆ Annf . This annihilator ideal shows an unexpected behaviour. The most striking case is when (n, d) = (9, 3). It turns out that dim C (Annf ) 3 = 4, see [LN]. At the same time a naive guess is that (Annf ) 3 is spanned by f only. Additionally, computer experiments suggest that (f ) and Annf agree in low degrees.
Problem P. Let f be a form of odd degree d in E. Is it true that (Ann(f )) i = (f ) i , for i < (n − d)/2?
We finish our list of problems with the following conjecture stated in [CLN], which connects the question about the Hilbert series of generic forms in the exterior algebra with the Hilbert series of power ideals in the commutative setting.
Conjecture 4.3. Let f and g be generic quadratic forms in E and let ℓ 1 and ℓ 2 be two generic linear forms in S. Then the Hilbert series of E/(f, g) is equal to the Hilbert series of S/(x 2 1 , . . . , x 2 n , ℓ 2 1 , ℓ 2 2 ) and is given by 1 + a(n, 1)t + a(n, 2)t 2 + · · · + a(n, s)t s + · · · , where a(n, s) is the number of lattice paths inside the rectangle (n + 2 − 2s) × (n + 2) starting from the bottom-left corner and ending at the top-right corner by using only moves of two types: either (x, y) → (x + 1, y + 1) or (x − 1, y + 1).