Papers

We study numerical semigroups whose defining ideals admit determinantal presentations by the $2\times2$ minors of a $2\times n$ matrix. A conjecture of Cuong, Kien, Matsuoka, and Truong relates such presentations to arithmetic progressions of pseudo-Frobenius numbers. We prove a numerical criterion for this determinantal presentation, reducing the conjecture to a problem on factorizations of maximal elements of an Apéry set in the numerical semigroup. As an application, we verify the conjecture in embedding dimension four under a unique factorization assumption on an Apéry set.

Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimension three, we obtain such ideals as inverse images $I_h=\varphi_H^{-1}(t^hk[t]\cap k[H])$. If $\ell$ is the least degree of a defining relation of $k[H]$, then $I_\ell$ is monomial, whereas $I_{\ell+1}$ has a binomial generator in the cases considered here. We determine $I_{\ell+1}$ for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal $I$ in these families, we prove that its Rees algebra is a Cohen–Macaulay normal domain. In the non-symmetric multiplicity-four case, $I_{\ell+1}$ is seven-generated; for $H=\langle4,9,15\rangle$, we prove that its Rees algebra is again a Cohen–Macaulay normal domain.

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_\Delta\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen–Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen–Macaulay if and only if $\Delta$ is a matroid, where loops are allowed. Thus, the Cohen–Macaulayness of a single nonexceptional window forces the Cohen–Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen–Macaulayness of $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen–Macaulay.

Understanding the relation between $\dim R$ and $\dim R[[x]]$ is a classical problem in commutative algebra. For a Noetherian ring $R$, one has $\dim R[[x]]=\dim R+1$, but the general case is considerably more delicate. In 1973, Arnold proved that finite power-series dimension requires the strong finite type (SFT) condition, whereas, in 2002, Coykendall constructed a one-dimensional SFT domain whose power series ring has infinite dimension. The question of Coykendall and Gilmer whether $\dim R[[x]]<\infty$ forces $\dim R[[x]]\le2\dim R+1$ was answered negatively by Kang and Park in 2009. In this paper, we prove that, as $R$ ranges over the nonzero commutative rings with identity, the finite pairs $(\dim R,\dim R[[x]])$ are exactly $(0,1)$ and the pairs $(n,m)$ with $1\le n<m$.

We study the geometry of parameter spaces of homogeneous ideals in k[x,y,z] minimally generated by three forms of the same degree, stratified by the degree of their base scheme and by the initial degree of their syzygies. We realize these strata through Quot and Hilbert schemes, construct universal families of Bourbaki schemes, and prove smoothness and irreducibility results for the free loci. We then compare these spaces with the loci of gradient ideals and plane curves. Using slope semistability of the associated syzygy bundles, we give a vector-bundle interpretation of the refined du Plessis–Wall bounds and obtain the same numerical restriction for arbitrary three-generated ideals. Finally, in the case of triples of cubic forms, we determine all nonempty degree strata, prove that they are smooth and irreducible, and identify their gradient loci in terms of the Bourbaki stratification of reduced quartic plane curves.

Let $G_{n,m}=\overline{\C_n^{[m]}}$, where $\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $\Delta(G_{n,m})$. We introduce a family $\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\in\mathcal{S}_V(k,m)$, the induced subcomplex $\Delta[W]$ has the homotopy type of $\mathbb{S}^1$, whereas for $W\notin\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.

We prove that there is a function $f:\mathbb N\to\mathbb N$ such that $\chi(\Gamma(R))\leq f(\omega(\Gamma(R)))$ for the zero-divisor graph of any ring $R$, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming $\chi(\Gamma(R))=\omega(\Gamma(R))$ for unital commutative rings. While previous counterexamples satisfy $\chi(\Gamma(R))\leq \omega(\Gamma(R))+2$, we obtain the lower bound $f\geq k^{\Omega(\log k)}$ among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are $\chi$-bounded.

We study the structure and distribution of atoms in the monoid $\mathcal{M}{\rm on}(R)$ of nonzero monomial ideals of the polynomial ring $R=K[X_1,\ldots, X_N]$, with $N\ge 2$. We characterize the atoms with at most four minimal generators and introduce the density $d(N,\mu)$ of atoms among monomial ideals with $\mu$ minimal generators. For $\mu\le 4$, we prove the existence of this density and determine its asymptotic behavior as $N\to\infty$. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of $\mathcal{M}{\rm on}(R)$ generated by equigenerated ideals.

A multiplicative set $S$ of a commutative ring $R$ is strongly multiplicative if every family $(s_i)_{i \in I}$ of elements of $S$ admits a common multiple in $S \cap \bigcap_{i \in I} s_iR$. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that $S$ is strongly multiplicative if and only if localization at $S$ commutes with arbitrary intersections of ideals, if and only if $R_S$ is the localization at an idempotent, and if and only if $D(S)$ is clopen in $\Spec(R)$. We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag–Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed–Malek question on the role of strong multiplicativity for chains of $S$-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for $R_S$, and connect the theory with the regular $m$-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.

An $A$-module $E$ is annihilator multiplication if the annihilator of each element equals that of $IE$ for a finitely generated ideal $I$. We characterize rings for which every faithful module has this property as the commutative quasi-Frobenius rings, and rings for which every module has the property as the Artinian principal ideal rings. Both characterizations reduce to two-generated modules. Over a reduced ring with finitely many minimal primes, faithful annihilator multiplication modules are exactly the regular-torsion-free modules, and principal ideals suffice in the definition. An ascending chain condition on annihilator ideals gives finite detection of the module annihilator, yielding localization and graph rigidity results. We also prove equality of associated primes with those of the faithful quotient and establish projective tensor and trace characterizations. Over local square-zero rings, the property is equivalent to nonsingularity of a bilinear multiplication map. A projective dimension argument gives a sharp length bound, attained by explicit faithful indecomposable nonprojective modules whose endomorphism rings are computed. Every cyclic submodule of these examples embeds in the ring, although the whole module is not torsionless. Support, hereditary torsion, and amalgamation criteria connect these structural results with further module constructions.

We study polynomial maps in three variables with nilpotent Jacobian over a field of characteristic zero. The proposed classification reduces maps with linearly independent components to a family determined by a univariate polynomial evaluated at a quadratic coordinate. The geometric part of the argument produces two algebraically dependent constant linear combinations of the components. A derivation argument then yields the normal form over the original field. We obtain explicit polynomial inverses and tame factorizations. We then study higher-dimensional maps in which all but the last three components depend on three variables. For this class, we give separate normal forms for the two three-variable blocks and deduce stable tameness from the residue-field criterion of Berson, van den Essen, and Wright.

We prove that a Noetherian integral domain is regular if, and only if, the derived pushforward of the structure sheaf along every blowup is a perfect complex. For suitable local rings, we give a construction which realizes a shift of the residue field as a direct summand of the derived pushforward of the structure sheaf along an explicit blowup.

We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2<K_Y^2-8\chi(\mathcal{O}_Y)=\tau(Y)$, the signature of $Y$ over $\mathbf{C}$, leaves no nonzero $\mathcal{E}$ with $H^1(Y,\mathcal{E}(tH))=0$ for all $t$. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of $\mathcal{E}$ being compared instead with the Horrocks splitting of its direct image on $\mathbb{P}^2$. Hirzebruch's Hesse surfaces with $H_n=4A_n-E_n$, $n\ge3$, qualify. In dimension $m$ the obstruction reads $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension $\ge3$, and the threefold examples show that the characteristic-$p$ hypothesis in the criterion of Shimomoto-Tavanfar is essential.

Let $K^h$ be a Henselization of a valued field $(K,v),$ $w$ a valuation-transcendental extension of $v$ to $K[x],$ and $\phi$ a key polynomial for $w$. In this paper, we prove that if $\phi$ is irreducible over $K^h,$ then the simple extension $L|K,$ generated by a root of any key polynomial for $w,$ is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for $L|K$ to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of $p$-regular generators for number fields.

We show that for any henselian valued field of mixed characteristic $(0,p)$, the (existential) $\mathcal{L}_{\mathrm{val}}$-theory of the valued field is determined by the (existential) theory of the value group in $\mathcal{L}_{\mathrm{oag}}$ with a constant for $v(p)$ and the (existential) theory of the residue ring $\mathcal{O}_v/(p)$ in an expansion $\mathcal{L}_{\mathrm{Witt}}$ of the language of rings, provided $\mathcal{O}_v/(p)$ is semi perfect. We moreover show that the $\mathcal{L}_{\mathrm{Witt}}$-structure on $\mathcal{O}_v/(p)$ is $\mathcal{L}_{\mathrm{ring}}$-definable using constants, and that this is exactly the structure induced on $\mathcal{O}_v/(p)$ by the ambient valued field. As a consequence, we obtain a relative quantifier elimination result (eliminating $K$-quantifiers) in a suitable language for the theory of roughly deeply ramified henselian valued fields of mixed characteristic $(0,p)$.

Let $R$ be a commutative ring with nonzero identity and let $R^\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\#)$. Its basic open sets are \[ B_a=\{[b]\in EC(R^\#): b\mid a^n \text{ for some }n\geq 1\}. \] The relation \[ [b]\in B_a \quad\Longleftrightarrow\quad \sqrt{aR}\subseteq\sqrt{bR} \] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.

An automorphism of the polynomial ring $k[x_1,\ldots ,x_n]$ over a field $k$ is said to be $\mathit{tame}$ if it can be obtained by composing affine automorphisms and elementary automorphisms, and $\mathit{wild}$ otherwise. Jung and van der Kulk showed that every automorphism of $k[x_1,x_2]$ is tame. In 1972, Nagata conjectured that a certain automorphism of $k[x_1,x_2,x_3]$ is wild. In 2003, Shestakov and Umirbaev proved this conjecture for $\mathop{\mathrm{char}}\nolimits k=0$. The purpose of this paper is to prove the conjecture for $\mathop{\mathrm{char}}\nolimits k\ge 7$. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.

For a commutative ring $R$, we call an $R$-module $B$ Baer if $\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $\tau_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes $\tau_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\varepsilon]/(\varepsilon^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not $\tau_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\EC(D)$-module is projective, whereas $\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not $\tau_q$-semisimple.

Let $R$ be a one-dimensional commutative Noetherian ring with a unique minimal prime $\mathfrak p$ such that $D=R/\mathfrak p$ is a principal ideal domain. We give a complete characterization of the iso-Artinian property in this class, that is, the following conditions are equivalent: $R$ is iso-Artinian; $\mathfrak pR_{\mathfrak p}=0$; $len_R(\mathfrak p)<\infty$; $\mathfrak p/\mathfrak p^2$ is torsion over $D$. As a consequence, we completely answer the Question~3.9 of Daneshvar and Divaani-Aazar: under their hypotheses $Min R\subsetneq Ass R$, the ring is iso-Artinian precisely when its nilradical has finite length, and it is non-iso-Artinian precisely when the conormal module has positive rank.

Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\sup\{depth R_{\mathfrak p}\mid \mathfrak p\in GSpec(R)\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\geq2$ and $h\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.

Let $R$ be an affine domain of dimension $d$ over an infinite field, with normalisation $\tilde{R}$ smooth, and let $\mathfrak{c}$ be the conductor. We answer a question of Nori in the affirmative when $R$ is singular only in high codimension.

This paper introduces a new polynomial model for the exact variant of the Syndrome Decoding Problem (SDP) in the binary case. The model is based on elementary symmetric polynomials. We estimate the computational complexity of solving the corresponding polynomial system by establishing bounds on the degree of regularity and on the solving degree of the ideal associated to the model. The complexity estimate is lower than for previous polynomial models. We also provide a variant of the model whose complexity depends directly on the specific instance of the SDP and is lower than for the first model. Finally, we discuss how to apply our approach to solve other variants of the SDP.

Let $T_{m,n}$ be the tadpole graph obtained by joining a cycle $C_m$ to a path $P_n$ by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs $(m,n)$ for which the Artinian algebra defined by the edge ideal of $T_{m,n}$ together with the squares of all variables has the weak Lefschetz property.

In the paper, we classify all polynomial maps of the form $H=(u(x,y),\allowbreak v(x,y,z),h(x,y,z))$ in the case that $JH$ is nilpotent and $\deg_zv\geq 2\deg_zh$. Then we give the structure of $H=(u(x,y),v(x,y,z),h(x,y,z))$ if $JH$ is nilpotent and $\deg_zh\leq 3$.

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.

We study relative global dimensions of extensions $B \subseteq A$ of Artin algebras over a perfect field. A controllable extension is introduced as one for which the relative global dimension is determined by the ordinary global dimension of the quotient $A/AJ(B)A$. General inequalities and sufficient conditions for controllability are established, including the case in which $J(B)$ is a two-sided ideal of $A$. The interaction between relative global dimensions and relative projective dimensions of the regular bimodule is then studied. The class of homologically controllable extensions is introduced, and it is shown to be closed under tensor products. Consequently, for homologically controllable extensions $B \subseteq A$ and $D \subseteq C$, we obtain \[\operatorname{gldim}(A \otimes C, B \otimes D)=\operatorname{gldim}(A,B) + \operatorname{gldim}(C,D).\] The proof relies on a relative Künneth-type result for tensor products of complexes, which may be of independent interest. Finally, controllable extensions with prescribed relative global dimension are constructed, along with families of non-controllable extensions for which the relative global dimension differs arbitrarily from the global dimension of $A/AJ(B)A$.

We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry. Over $\mathbb{C}$, a width-$r$ network with activation $z\mapsto z^d$ computes a sum of $r$ $d$-th powers of linear forms, whose Zariski closure is a Veronese secant variety. Ideal elimination therefore yields polynomial certificates of nonrepresentability. We implement this construction as a generic architecture-to-certificate pipeline. For quadratics, we recover the exact symmetric determinantal description and explain its dimension through orthogonal symmetry. In higher degree, the implementation recovers classical catalecticant and secant equations and maps the practical reach of direct elimination across a finite architecture sweep. We also derive the exact population loss floor for a rank-two quadratic network on the sphere, illustrating how an algebraic obstruction induces irreducible approximation error.

We develop a descent theory for sparse factorization-profile images, asking which functions on a normalization descend to the actual image. Whole-fiber completions yield a Fourier-zero classification for homogeneous masks, with a sharp stable converse, and a Fourier-line classification for mixed aperiodic profiles in the stated marked stable ranges. For homogeneous profiles, bounded integer relations determine geometric fibers and seminormalization, while explicit semigroup models compute actual image algebras, conductors, and depth in substantial families. For singleton–complement profiles, exact one-jet conditions determine the finite-output algebra, conductor, and projective descent through the first nonnormal boundary; the local type is computed in the smooth-boundary range and at the total-zero point of the first nonnormal layer. We also prove a saturated root-of-unity specialization theorem and give separating examples showing that normalization, natural polarizations, point fibers, seminormalization, and even the source conductor need not determine the actual image.

In this article, we compute the Hilbert series of the generalized binomial edge ideal associated with the join of two graphs. Consequently, we compute the Hilbert series of complete $r$-partite graphs. Further, we show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal. This provides a reduction technique for computing them. As an application, we compute the dimension and multiplicity of several classes of graphs.

We prove the Eisenbud–Ulrich conjecture on the powers of linearly presented ideals in characteristic zero. More generally, we study stabilization bounds for powers of m-primary ideals in a polynomial ring with partially linear resolutions, making further progress toward the more general Eisenbud–Huneke–Ulrich conjecture. We prove a global generation result on relevant higher syzygy bundles, which is a crucial ingredient to the proof of the Eisenbud–Ulrich conjecture. We also establish lower bounds for the number of minimal generators in two settings: ideals generated by quadrics and ideals that are virtually linearly presented. Motivated by these results, we formulate a conjecture predicting sharp lower bounds for the number of minimal generators of such ideals.