Fröberg's conjecture: the Hilbert–Poincaré series of ideals generated by generic homogeneous forms

Let $K$ be an infinite field, $R=K[x_1,\dots,x_n]$ with the standard grading, and let $f_1,\dots,f_m$ be generic homogeneous polynomials of degrees $d_1,\dots,d_m$. Fröberg's conjecture (1985) predicts that the Hilbert–Poincaré series of $R/\langle f_1,\dots,f_m\rangle$ equals the truncated rational form $\left\lfloor\prod_{i=1}^{m}(1-z^{d_i})/(1-z)^n\right\rfloor$, i.e. the power series obtained from $\prod_{i}(1-z^{d_i})/(1-z)^n$ by keeping its initial run of coefficients while they are positive and setting all later coefficients to zero. The conjecture is equivalent to Pardue's Conjecture B (generic homogeneous sequences over an infinite field are semi-regular) and is implied by the stronger Moreno–Socías conjecture on generic initial ideals being almost reverse lexicographic. It is open in general: only restricted cases (complete intersections, $n\le 3$, $m=n+1$ in characteristic zero, several equal-degree families, and recent partial regimes) are proved. All three conjectures remain open.

Background

For a graded polynomial ring $R=K[x_1,\dots,x_n]$ over a field $K$, the Hilbert–Poincaré series of a graded quotient algebra $A=R/I$ is the power series $\mathrm{HS}_{A}(z)=\sum_{t\ge 0}\dim_K A_t\, z^t$ encoding the dimensions of the graded pieces; it is the standard invariant controlling the size and complexity of $A$. Call homogeneous polynomials $f_1,\dots,f_m$ of degrees $d_1,\dots,d_m$ generic if their coefficients are algebraically independent over $K$; a property of such systems holds generically if it holds on a nonempty Zariski-open set of the coefficient space. Since algebraically independent coefficients cannot be written down over a small field, genericity is usually studied via this Zariski-open formulation, and the case of finite fields is handled by separate transfer arguments.

A sequence $f_1,\dots,f_m$ is called semi-regular if each $f_i$ acts with maximal rank on what precedes it: for every $t$, the multiplication map by $f_i$ from the degree-$(t-d_i)$ piece of $R/\langle f_1,\dots,f_{i-1}\rangle$ to its degree-$t$ piece is injective or surjective, whichever of the two is possible given the two dimensions (it fails to be injective exactly when the quotient by the first $i-1$ forms has no room for a new non-zerodivisor in degree $t$). Semi-regularity forces the Hilbert–Poincaré series of the quotient to equal a purely combinatorial formula: the truncated rational function $\left\lfloor\prod_{i=1}^{m}(1-z^{d_i})/(1-z)^n\right\rfloor$. Here the truncation is the operation on power series that keeps the coefficient of $z^t$ equal to the corresponding coefficient $a_t$ of the expansion of $\prod_{i}(1-z^{d_i})/(1-z)^n$ as long as $a_0,a_1,\dots,a_t$ are all positive, and sets it to zero otherwise; equivalently, the series is cut after its last consecutive positive coefficient. Fröberg conjectured in 1985 that this truncated form is the Hilbert–Poincaré series of the quotient by a generic sequence, i.e. that no unexpected cancellation or syzygy occurs generically (DOI 10.7146/math.scand.a-12092). Pardue later formulated the equivalent Conjecture B — generic homogeneous sequences over an infinite field are semi-regular (DOI 10.1016/j.jalgebra.2010.04.018) — and proved the equivalence of the two statements.

For a fixed degree sequence the problem has a useful structural backbone. In each degree $t$ the Hilbert function of the quotient is $\dim_K R_t$ minus the rank of an explicit linear map $\bigoplus_i R_{t-d_i}\to R_t$ (multiply by the $f_i$ and add), whose matrix entries are linear in the coefficients of the forms; rank is lower semicontinuous, so the pointwise minimum of the Hilbert function over all systems of the given degrees is attained on a nonempty Zariski-open set and equals the generic Hilbert function — the generic Hilbert series is the coefficientwise minimum, a fact stressed by Fröberg–Lundqvist (arXiv:1711.01232), who note it is 'truly minimal', not merely minimal in the lexicographic sense. Fröberg's 1985 paper also proved an inequality for Hilbert series of graded algebras implying that the Hilbert series of any quotient by forms of the given degrees is at least the truncated form (in the lexicographic comparison used there); this is why exhibiting a single system that attains the truncated form proves the conjecture for that degree sequence, the standard technique behind most known cases. Two consequences follow: a single explicit system whose Hilbert function falls below the truncated form in some degree would refute the conjecture outright, while a special system whose Hilbert function merely lies above it refutes nothing.

A stronger prediction, due to Moreno–Socías, concerns generic initial ideals. The initial ideal of $I$ with respect to a term order is the monomial ideal generated by the leading monomials of all elements of $I$; for the degree-reverse-lexicographic order and generic coordinates the initial ideal $\operatorname{gin}(I)$ is Borel-fixed (stable under a natural group action), and it determines the Hilbert–Poincaré series. The Moreno–Socías conjecture asserts that for a generic sequence the degree-reverse-lexicographic generic initial ideal is almost reverse lexicographic (also called weakly reverse lexicographic): whenever a monomial $x^\mu$ is a minimal generator of it, every monomial of the same total degree that is larger in the degree-reverse-lexicographic monomial order also belongs to it. This property forces the Hilbert series to take the truncated form above, so the conjecture implies both Fröberg's and Pardue's statements (the implication is due to Pardue).

The question matters beyond pure commutative algebra because the semi-regularity assumption is the standard heuristic model for random overdetermined polynomial systems: Gröbner-basis solvers (such as the F4/F5 algorithms, which repeatedly linearize the system degree by degree) are predicted to terminate at a degree read off from the truncated rational form, and these predictions underpin the security estimates of multivariate public-key cryptosystems. The solving degree of a system (the maximal degree reached during Gröbner-basis computation) is known by explicit examples not to be bounded, in general, by the degree of regularity (the degree at which a semi-regular sequence exhausts the quotient) or by the first fall degree (the first degree at which an unexpected low-degree algebraic consequence appears), as documented by Bigdeli–De Negri–Muzika Dizdarevic–Gorla–Minko–Tsakou (arXiv:2011.01032) and Caminata–Gorla (DOI 10.1016/j.jsc.2022.05.001). This is what makes the conjectural generic behavior, rather than a proven theorem, the load-bearing ingredient of the complexity predictions.

Pardue's Conjecture B, the equivalent Fröberg conjecture, the stronger Moreno–Socías conjecture, and the open problem of tight provable solving-degree estimates are unresolved (Kudo–Yokoyama, arXiv:2401.07768). The three conjectures form one nested core with a single scientific objective (determine the generic Hilbert–Poincaré series, equivalently the generic initial-ideal structure).

Problem Statement

Let $K$ be an infinite field, $R=K[x_1,\dots,x_n]$ with the standard grading, and let $f_1,\dots,f_m$ be generic homogeneous polynomials of degrees $d_1,\dots,d_m$ (all $d_i\ge 1$). Prove or refute Fröberg's conjecture: the Hilbert–Poincaré series of $R/\langle f_1,\dots,f_m\rangle$ equals

$\left\lfloor\frac{\prod_{i=1}^{m}(1-z^{d_i})}{(1-z)^n}\right\rfloor,$

the truncation of the power series expansion of $\prod_{i=1}^{m}(1-z^{d_i})/(1-z)^n$ obtained by keeping each coefficient as long as all coefficients up to it are positive and replacing all later coefficients by zero. The assertion must hold (or fail) uniformly for all numbers of variables $n$, all numbers of forms $m$, and all degree sequences $(d_1,\dots,d_m)$ over an arbitrary infinite field. Equivalently (Pardue's Conjecture B): every generic homogeneous sequence over an infinite field is semi-regular. The conjecture is implied by the stronger Moreno–Socías conjecture — that the generic initial ideal with respect to the degree-reverse-lexicographic order is almost reverse lexicographic — which implies both equivalent statements. No restriction on $n$, $m$, the degrees, or the characteristic is permitted: partial regimes beyond the known cases listed in Background do not settle the problem as stated. A refutation must establish deviation for the generic system, not merely for special systems: the generic Hilbert function is the coefficientwise minimum over all systems of the given degrees, so a single explicit system whose Hilbert function falls below the truncated form in some degree would refute the conjecture, while a special system whose Hilbert function merely lies above it does not by itself.

References

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