Papers

We introduce invariants we call $k$-divisibility and $d$-dominance, which are combinatorial constraints on the minimal generators of monomial ideals. We then prove that $k$ (respectively, $d$) is an upper bound (respectively, lower bound) for the projective dimension of a $k$-divisible (respectively, $d$-dominant) monomial ideal. This leads us to an explicit minimal resolution of $k$-divisible, $d$-dominant ideals when $d=k-1$ via a Lyubeznik resolution. This framework allows us to prove a conjecture made by Alesandroni. Using these invariants, we also provide a characterization of monomial ideals whose minimal free resolution is attainable from the Taylor resolution by performing exactly one consecutive cancellation. We also prove that for all $q,k,d$ in the range $1\leq d<k \leq q-1$, there exist monomial ideals that are $k$-divisible and $d$-dominant with $q$ minimal generators.

This paper extends Macdonald's duality theory and Cuong-Nam's local homology theory – which is in some sense dual to the local cohomology theory of A. Grothendieck [Gro1] – to semilocal noetherian rings equipped with the $J$-adic topology. By overcoming the key structural obstructions of the semilocal setting via Zöschinger's structure theory and semilocal Matlis duality, this work establishes topological limit duality isomorphisms and links local homology with local cohomology. Ultimately, this framework provides the foundational machinery required to extend the classical results of Macdonald [Mac] and Cuong-Nam [C-N1] to the semilocal setting.

We consider the big finitistic projective and flat dimensions of a commutative ring and of its formal power series ring. When the ring is Noetherian, both invariants increase by one. Coherence of the power series ring does not force either equality to hold. For every \(n\ge0\) and every \(d\) with \(n+1\le d\le\infty\), the flat pair \((n,d)\) can be realized. The finite projective pairs are realized when \(n\ge1\), and also for \((0,1)\). For every \(n\ge0\), one family realizes \((n,\infty)\) for both invariants at the same time. The seed is a zero-dimensional perfect local ring. Its power series ring has countably presented modules whose flat dimension is \(r\) and whose projective dimension lies between \(r\) and \(r+1\), for every \(r\ge1\). The remaining finite projective case is controlled by a dichotomy for perfect coefficient rings. If the Jacobson radical is nilpotent, the power series value is one; if it is not nilpotent, there is a module of projective dimension exactly three. In particular, the pair \((0,2)\) is impossible. For one fixed countable perfect ring, the power series ring has finitistic flat dimension one. Its finitistic projective dimension is \(m+2\) when \(2^{\aleph_0}=\aleph_m\) with \(1\le m<\infty\), and is infinite when \(2^{\aleph_0}>\aleph_\omega\).

For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen–Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that $\mathcal{GI}(\mathrm{QCoh}(X))$ is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring $S$ such that $\mathcal{GI}(S_{\mathfrak{p}})$ forms a covering class for all prime ideals $\mathfrak{p}$, but $\mathcal{GI}(S)$ do not form a covering class.

In this paper, we improve the upper bound for the tensor rank of the $n \times n$ determinant tensor to $\mathcal{B}_n-2^{n-1}+n+1$ by finding a new tensor decomposition of the tensor, where $\mathcal{B}_n$ denotes the $n$-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston–Goucher–Johnston with a tensor that is essentially the same as the $n \times n$ permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the $n \times n$ determinant polynomial for $n \geq 17$ as well.

The complete intersection forms are the homogeneous polynomials whose annihilator ideal or the Macaulay dual, is an Artinian complete intersection ideal. In this paper, we compute the Waring rank of certain complete intersection forms. As an application, we compute the Waring rank of binomials whose annihilator is a complete intersection ideal. Further, we also compute the Waring rank of the product of binary forms in different sets of variables.

We study the local $\mathrm v$-numbers of binomial edge ideals from a combinatorial perspective. For each minimal prime $P_{\mathcal S}$ of $J_G$, indexed by a cutset $\mathcal S$ of $G$, we consider two combinatorial invariants, $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, defined in terms of transversals of suitable hypergraphs, and show that the local $\mathrm v$-number $\mathrm v_{P_{\mathcal S}}(J_G)$ satisfies $\mathrm v^-_{\mathcal S}(G)\leq \mathrm v_{P_{\mathcal S}}(J_G)\leq \mathrm v^+_{\mathcal S}(G)$. Under a suitable hypothesis on the connected components, we provide explicit graph-theoretic descriptions of these invariants and analyze when they coincide, giving a combinatorial way to determine the local $\mathrm v$-numbers. As an application, we derive a sufficient condition for this coincidence, which applies to a broad class of graphs. Furthermore, we identify a constant that controls the gap between $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, thereby showing that the local $\mathrm{v}$-number lies within a bounded interval. Applications to cycles and trees illustrate both the equality criterion and the resulting bounds.

We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the $3\times3$ minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of $\operatorname{Ext}$ modules, and introduce the dual notion of the cotangency flattening, related to dimensions of $\operatorname{Tor}$ modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.

For a $d$-dimensional Noetherian local ring $(R,\mathfrak{m})$, with $d>0$, let $c_{\infty}(R)$ be the limit, as $N\to\infty$, of the suprema of $e_R(I)/(d!\ell_R(R/I))$ over ideals $I$ with $\sqrt{I}=\mathfrak{m}$ and $\ell_R(R/I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and Cohen–Macaulay on the punctured spectrum, with $d\geq2$ and infinite residue field, then every nonzerodivisor $x\in\mathfrak{m}$ satisfies \[ c_{\infty}(R)\leq \max\left\{c_{\infty}(R/xR), \sup_{\mathfrak{p}\ne\mathfrak{m}}e(R_{\mathfrak{p}})\right\}. \] As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $c_{\infty}(R)=1$ whenever $\widehat{R}$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech–Mumford constant $c_{\mathrm{LM}}(R)$ is attained when $\widehat{R}$ has isolated singularities and $c_{\mathrm{LM}}(R)>1$. For reduced equidimensional Cohen–Macaulay standard graded algebras of dimension $\geq2$ over infinite perfect fields, we prove attainment and rationality of $c_{\mathrm{LM}}$ for the completed vertex local rings of sufficiently high Veronese subrings. In characteristic zero, a refinement of our method produces nonisolated klt affine cones in every dimension $d\geq3$, with $d-2$-dimensional singular loci, whose completed vertex local rings have $c_{\infty}=1$ and arbitrarily large $c_{\mathrm{LM}}$.

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

We study the Solomon–Terao algebra and the Solomon–Terao complex of a central hyperplane arrangement $\mathcal A$ in $\mathbb C^\ell$. Let $\eta$ be a homogeneous polynomial of degree $d\geq2$ whose restriction to every flat of $\mathcal A$ has no critical point away from the origin. We prove that the Solomon–Terao algebra $\mathrm{ST}(\mathcal A,\eta)$ has top degree $|\mathcal A|+\ell(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{ST}(\mathcal A,\eta)$ is bounded below by that of the Orlik–Solomon algebra. Both results hold without assuming that the arrangement $\mathcal A$ is tame. Finally, we prove that $\mathcal A$ is tame if and only if all positive-degree homology groups of the Solomon–Terao complex associated with $\eta$ vanish.

Let $R$ be an integral domain and let $w$ denote the $w$-operation on $R$. A nonzero GV-torsion-free $R$-module $M$ is called $w$-secondary if, for every $r\in R$, either $M/rM$ is GV-torsion or $r^nM=0$ for some $n\in\N$. A $w$-module $M$ is said to admit a $w$-secondary representation if $M=(N_1+\cdots+N_n)_w$, where each $N_i$ is a $w$-secondary submodule of $M$. Such a representation is called minimal if the ideals $P_i=\sqrt{\Ann_R(N_i)}$ are pairwise distinct, and $M$ is called $w$-representable if it admits such a representation. If the sum can be chosen direct, then $M$ is called strongly $w$-representable. We characterize $w$-secondary modules and investigate their localization and $w$-closure properties. For a nonfield domain $R$, we prove that $R$ is $t$-almost Dedekind if and only if $\wdim R=1$ and every $w$-representable $w$-module is locally strongly $w$-representable: each of its nonzero localizations at maximal $w$-ideals is strongly representable. Global strong $w$-representability is retained when all the radical annihilators in the representation are nonzero.

Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on $n$, using a filtration of the kernel bundle of $T_{n,L}$ to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when $g\geq1$ and $n\geq2$, $T_{n,L}$ satisfies property $(N_p)$ for $\mathrm{deg}(L)\geq(n+1)g+p$. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.

In this paper, introduce a new finiteness condition for commutative rings, called $Z$-Noetherian, under which every ideal contained in the set of zero-divisors $Z(R)$ is finitely generated. This condition lies properly between Noetherian and $Nil_{*}$-Noetherian. We establish intrinsic characterizations of $Z$-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that $Z(R)$ is an ideal, we obtain a primary decomposition theorem for ideals contained in $Z(R)$. For $ZD$-rings, we prove a Cohen-type criterion characterizing $Z$-Noetherian in terms of the finite generation of prime ideals contained in $Z(R)$. We further establish $Z$-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for $Z$-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of $Z$-Noetherian rings.

It is shown that a certain finite partially ordered set arising from an algebra with straightening laws on a finite distributive lattice satisfies the $\lambda$-chain condition. In addition, its application to commutative algebra is briefly discussed.

We prove the geometric halves of Hilbert's Nullstellensatz and of the real Nullstellensatz for submodules of finitely generated modules and for coherent sheaves, complementing our earlier result that semiprime (resp. real) submodules are intersections of prime (resp. real prime) submodules. First, we pass from prime to maximal submodules: over a Jacobson ring, a prime submodule $P$ of $M$ is an intersection of maximal submodules whenever $M/P$ is a torsionless $R/(P:M)$-module, in particular whenever $M/P$ is finitely generated (the latter case is due to Maani Shirazi and Sharif). Consequently, if $R$ is a finitely generated algebra over an algebraically closed field and $M$ is finitely generated, the smallest semiprime submodule containing a submodule $N$ of $M$ consists of all $f\in M$ such that, for every point $a$, the value $f(a)$ (i.e. the image of $f$ in the fibre $M/\mathfrak{m}_aM$) lies in $N(a)$ (i.e. the image of $N$ in $M/\mathfrak{m}_aM$). We prove the real analogues over real Jacobson rings; for finitely generated algebras over a real closed field they are determined by the real points. The module classifications and directional vanishing formulas yield both Nullstellensätze for coherent sheaves on schemes of finite type over a field. We also prove compatibility of semiprimeness, reality and both radicals with arbitrary localization, and the resulting stalk criteria.

Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\in(N:m)M$ implies $m\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an arbitrary $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\notin\p Q$ for all $s\notin\p$, where $\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\sum_i m_i\otimes m_i=0$ in $M\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.

We prove that a commutative integral domain is Gorenstein Prüfer if and only if every overring is Gorenstein flat over the original domain. It is enough to test overrings that are finitely generated as algebras. The proof combines finite duality with a conductor-quotient argument. Under the overring hypothesis, the conductor quotient of a proper overring generated by two elements as a module is an IF-ring, and degree-one duality lifts first syzygies of finitely presented quotient modules to totally reflexive modules. This gives a bound of one for the Gorenstein flat dimension of every module annihilated elementwise by conductor powers. Quadratic integral elements provide such conductors at nonnormal localizations, while normal localizations give principal-ideal neighborhoods. Constructible compactness yields a finite cover, and a bounded complex of localizations combines these bounds over the original domain. Consequently every finitely generated ideal is Gorenstein flat and the domain is coherent. Examples distinguish finite-type from module-finite tests and Gorenstein flatness from ordinary flatness.

There exists a one-dimensional local Gorenstein Prüfer domain whose integral closure is not Prüfer. We prove that the recently constructed coherent counterexample to the normalization problem has this stronger Gorenstein property. The argument establishes permanence of divisoriality of finitely generated ideals under ultraproducts and quotients by divided primes. It also supplies uniform matrix factorizations: in an ultraproduct of plane-curve domains, every nonzero ideal generated by $r$ elements is a cycle of a two-periodic totally acyclic complex with free terms of rank $r$. The divided prime defining the one-dimensional example is flat, so these complexes descend through the quotient and localization. A specified two-generated ideal remains noninvertible in every integral overring. Over the normalization, its associated Laurent overring has nonzero first Tor with every injective cogenerator, although both rings are Gorenstein flat over the original domain. Examples distinguish this failure from a nonnormal Gorenstein curve with Prüfer normalization and show that the periodicity conclusion is stronger than the Gorenstein Prüfer property alone.

Let $\mathbb{F}_q$ be a finite field of any characteristic and $\rm GL(n,\mathbb{F}_q)$ the general linear group over $\mathbb{F}_q$. Suppose $W$ denotes the standard representation of $\rm GL(n,\mathbb{F}_q)$ and $\rm GL(n,\mathbb{F}_q)$ acts diagonally on the direct sum of $W$ and its dual space $W^*$. Let $G$ be any subgroup of $\rm GL(n,\mathbb{F}_q)$. Suppose the invariant field $\mathbb{F}_q(W)^G=\mathbb{F}_q(f_1,f_2,\dots,f_k)$, where $f_1,f_2,\dots,f_k\in\mathbb{F}_q[W]^G$ are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials $\ell_1,\ell_2,\dots,\ell_{n}$ in the invariant ring $\mathbb{F}_q[W\oplus W^*]^{G}$ such that the invariant field $\mathbb{F}_q(W\oplus W^*)^{G}$ is generated by $\{f_1,f_2,\dots,f_k,\ell_1,\ell_2,\dots,\ell_{n}\}$ over $\mathbb{F}_q$.

We give an ideal in a polynomial ring in three variables, generated by three elements and having height equal to three, but for which no permutation of its generators is a regular sequence. We prove that in a polynomial ring over an infinite field, an ideal of height n generated by n polynomials, always admits a set of n generators, for which any permutation is a regular sequence.

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $I\subseteq \mathfrak{m}$ be an ideal in $A$. Let $\ell(I)$ be the analytic spread of $I$. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth} \ A/I^n \geq t - \ell(I)$ for all $n\geq 1$.

We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing $(\mathcal{A}, \mathcal{B}, \mathcal{C})$: from $\mathcal{A}$ and $\mathcal{C}$ to $\mathcal{B}$, and conversely from $\mathcal{B}$ to $\mathcal{A}$ or $\mathcal{C}$. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenström's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.

Let $T$ be the theory of the valued field of complex Laurent series $\mathbb{C}(\!(t)\!)$ in the language of rings expanded by a constant for $t$ and by predicates for $n$-th powers. We introduce a $t$-adic analogue of Kochen's operator, namely $\gamma(x)=x/(1+tx^{2})$, and the associated Kochen ring $\Lambda_{L}$ of a field extension $L$ of a model $K$ of $T$. Our main result characterizes the field extensions of $K$ which embed into a model of $T$ (the formally $T$-adic} fields): $L$ is formally $T$-adic if and only if $1/t\notin\Lambda_{L}$. As an application we prove a Nullstellensatz for $\mathbb{C}(\!(t)\!)$: a polynomial $f\in K[X_{1},\dots,X_{n}]$ vanishes on the zero set of an ideal $I$ if and only if $f^{m}=\sum_{i}\omega_{i}f_{i}$ for some $m\geqslant 1$, generators $f_{i}$ of $I$ and coefficients $\omega_{i}$ in the ring $\Lambda_{K(X)}\cdot K[X]$. We derive the resulting correspondence between $R$-ideals and algebraic sets, and we compare natural incarnations of the associated $T$-spectrum: prime ideals, morphisms into models of $T$, and the space of types of $T$.

In this paper, we explore mixed Buchsbaum–Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum–Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form $IF$ for $\mathfrak m$-primary ideals $I$. The general converse problem in dimension at least two is stated at the end.

Let $R$ be a Noetherian regular local ring of mixed characteristic and dimension four, and let $r$ be a member of a regular system of parameters. We prove that every finitely generated projective module over $R[1/r]$ is free. We also prove that $\mathrm{Sp}_4(R[1/r])$ acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor–Witt sheaf and the alternating Witt group of $R[1/r]$. We establish these results using integral degree-two algebraic $K$-theory, supported hermitian $K$-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor–Witt multiplication and yields the required symplectic frames.

We reprove and generalize a result of Derksen describing the Hilbert polynomial of the vanishing ideal of a transversal subspace arrangement in projective space. When the subspaces have all codimension 2, we describe explicitly the local cohomology modules of the corresponding defining ideal. This is achieved via a complex naturally associated to the subspace arrangement, whose basic properties are also investigated.

Let $D_{m,n}\subset\C^{m\times n}$ be the union of the zero loci of the maximal minors of a generic $m\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\cong\GL_m(\C)\times\Gr(m,n)^\circ$, where $\Gr(m,n)^\circ$ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For $m\in\{2,n-2\}$ this gives closed formulas for the topology of $U_{m,n}$ and recovers results of Yim. Our main result, which is computer-assisted, is that $D_{3,5}$ is not a free divisor. This answers a question of Yim. We determine $\Der(-\log D_{3,5})$ explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in $\C^3$. Computations in further cases support the conjecture that $D_{m,n}$ is free only for $m\le 2$ or $n=m+1$.

We study roots of Bernstein-Sato polynomials of ideals. Our first main result gives a general partial differential equation criterion for detecting roots of the Bernstein-Sato polynomial of an ideal. For homogeneous case, we recast this criterion as an analytic condition, connecting the algebraic problem of locating roots with analytic properties of the defining data. As an application, we give a series of roots for the general determinantal ideal prove the strong monodromy conjecture for them. Keywords. Bernstein-Sato polynomial, determinantal variety, monodromy conjecture.

Let $A_n=\mathbb{k}[x_1^{\pm1},\ldots,x_n^{\pm1}]$ be a Laurent polynomial algebra over a field $\mathbb{k}$ of characteristic zero. We determine explicitly the automorphisms commuting with each locally finite derivation \[ D_\mu=\sum_{i=1}^n\mu_i x_i\frac{\partial}{\partial x_i}, \qquad \mu=(\mu_1,\ldots,\mu_n)^T\in\Bbbk^n. \] These are precisely the automorphisms $\rho(x_i)=\lambda_i x^{m_i}$, where $\lambda_i\in\mathbb{k}^\ast$, the exponent vectors $m_i$ are the columns of a matrix $M_\rho\in\operatorname{GL}_n(\mathbb Z)$, and $\mu^TM_\rho=\mu^T$. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.