Harm Derksen
Suppose that G is a linearly reductive group. We study the minimal free resolution of the invariant ring. If G is a finite linearly reductive group, then the ring of invariants is generated in degree at most |G|, the group order. We prove that the ideal of relations of a minimal set of generators of the invariant ring is generated in degree at most 2|G|.
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We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.
We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.
We show that a small variant of the methods used by Voisin in her study of canonical curves leads to a surprisingly quick proof of the gonality conjecture of Green and the second author, asserting that one can read off the gonality of a curve C from its resolution in the embedding defined by any one line bundle of sufficiently large degree. More generally, we establish a necessary and sufficient condition for the asymptotic vanishing of the weight one syzygies of the module associated to an arbitrary line bundle on C.
This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results a…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend previous results in the literature.
In a recent paper by L. Fel two new identities for the degree of syzygies are given. We present an algebraic proof of them, using only basic homological algebra tools. We also extend these results.
Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.
Syzygies of some invariant rings with cyclic group actions
本文研究循环群在二元多项式环上的作用及相应不变环的极小自由分解，完全分类了定义理想余维数为2和生成不变量多重度位于两直线并集上的情形。
Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $β_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.
Linear syzygies of curves in weighted projective space
本文将Green的$N_p$条件推广到加权射影空间，证明曲线高次嵌入满足这些条件，并在广泛情形下建立了正性与低次虚拟合冲之间的联系。
We study the minimal free resolution of the Veronese modules of the polynomial ring in n variables, by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We characterize when they are Cohen-Macaulay and we give a sufficient condition for the linearity of their minimal free resolution. We also conjecture that in 2 variables the Veronese modules have always pure resolutions. In addition, we give a closed formula for their Hilbert series. As an application of our results, we calculate the complete Betti diagrams of the Veronese subrin…
This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleadin…
For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $< d+i$ for $0\le i\le p$ (see \cite{EGHP2} for details). Much attention has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}). However, not very much is actually known about the case satisfying property $\textbf{N}_{3,p}$. In this paper, we give a sharp upper…
A minimal system of generators of the algebra of the centro-affine covariants for homogeneous planar cubic differential systems with linear part is known. With the help of the Gurevich theorem avoiding the Aronhold's identities based on the calculation of determinants, we describe the algebra of the centro-affne covariants for homogeneous planar cubic differential systems with free terms.
We derive polynomial identities of arbitrary degree $n$ for syzygies degrees of numerical semigroups S_m=<d_1,...,d_m> and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number g_m=B_m-m+1 of algebraically independent genera G_r and equations, related any of g_m+1 genera, where B_m=\sum_{k=1}^{m-1}β_k and β_k denote the total and partial Betti numbers of non-symmetric semigroups. The number g_m is strongly dependent on symmetry of S_m and decreases for symmetric semigroups and complete intersections.
We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.
We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series H(d^m;z) of nonsymmetric numerical semigroups S(d^m) of arbitrary generating set of positive integers d^m={d_1,...,d_m}, m\geq 3. These identities were obtained by studying together the rational representation of the Hilbert series H(d^m;z) and the quasipolynomial representation of the Sylvester waves in the restricted partition function W(s,d^m). In the cases of symmetric semigroups and complete intersections these identities become more compact.
A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.
It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between zero-sum theory and invariant theory is reversed. First it is shown how a presentation by generators and relations of the ring of invariants of an abelian group acting linearly on a finite dimensional vector space can be obtained from a presentation of the ring of invariants for the corresponding multiplicity free representation. This combined with a known deg…
We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.
The problem of finding generators of the $GL$-ideal of the relations between the generators of the algebra of invariants of the dihedral group acting on $m$-tuples of vectors from its defining $2$-dimensional representation is studied. It is shown that this $GL$-ideal is generated by relations depending on no more than $3$ vector variables. A minimal $GL$-ideal generating system is found for the case when $m=2$, and for the case of the dihedral group of order $8$ and arbitrary $m$.
We consider the following question: if a simplicial complex $Δ$ has $d$-homology, then does the corresponding $d$-cycle always induce cycles of smaller dimension that are not boundaries in $Δ$? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree.
We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.
A presentation by generators and relations of the $n$th symmetric power $B$ of a commutative algebra $A$ over a field of characteristic zero or greater than $n$ is given. This is applied to get information on a minimal homogeneous generating system of $B$ (in the graded case). The known result that in characteristic zero the algebra $B$ is isomorphic to the coordinate ring of the scheme of $n$-dimensional representations of $A$ is also recovered. The special case when $A$ is the two-variable polynomial algebra and $n=3$ is applied to find generators and relations of an algebra of invariants of…
The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. …
The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\ma…
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.
First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …
Over a field of characteristic $0$, we prove that for each $r \geq 0$ there exists a constant $C(r)$ so that the prime ideal of the $r$th secant variety of any Plücker-embedded Grassmannian ${\bf Gr}(d,n)$ is generated by polynomials of degree at most $C(r)$, where $C(r)$ is independent of $d$ and $n$. This bounded generation ultimately reduces to proving a poset is noetherian, we develop a new method to do this. We then translate the structure we develop to the language of functor categories to prove the $i$th syzygy module of the coordinate ring of the $r$th secant variety of any Plücker-emb…
We show that if C is a linearly normal smooth curve embedded by a line bundle of degree at least 2g+3+p then the secant variety to the curve satisfies N_{3,p}.
The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.
Let $F$ be a finite set of monomials of the same degree $d\geq 2$ in a polynomial ring $R=k[x_1,...,x_n]$ over an arbitrary field $k$. We give some necessary and/or sufficient conditions for the birationality of the ring extension $k[F]\subset R^{(d)}$, where $R^{(d)}$ is the $d${\it th} Veronese subring of $R$. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theoretic birationality criterion in characteristic zero
The Burch index is a new invariant of a local ring $R$ whose positivity implies a kind of linearity in resolutions of $R$-modules. For example, if $R$ has depth zero and Burch index at least $2$, then any non-free 7th $R$-syzygy contains the residue field as a direct summand. We also compute the Burch index in various cases of interest.
We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves ${\mathcal C} \subset \mathbb P^r_K$ of maximal regularity with $°{\mathcal C} \leq 2r -3.$ In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces $X \subset \mathbb P^r_K$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for "an …
Weighted Syzygies of Pointed Curves
针对亏格g光滑射影曲线上的点P，研究非次数1生成的截面环R_d的合冲，界定其生成元次数、Betti表支撑界，证明满足加权N_p条件的d阈值，给出Betti数确定的充分条件并得普通点R_{g+1}分解为纯的。
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebrai…
Resonance, syzygies, and rank-$3$ Ulrich bundles on the del Pezzo threefold $V_5$
本文研究了在del Pezzo三维流形V_5上，通过乌里奇向量包的截面零点描述亏格14、次数18的曲线的几何条件，并探讨了其与特殊秩2向量包的共振关系，以及曲线唯一性结论。
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.
Syzygies of Isotropic Kalman Varieties
本文引入辛向量空间上的各向同性卡尔曼簇，研究其定义方程、几何不变量和奇点，并推广到正交情形。
Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_n\leq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_i$ in the minimal resolution holds for $i \geq h-1$, i.e, we show that $T_i \leq T_a+T_{i-a}$ for $i\geq h-1$.
Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix entries. It is proven by Raicu and Weyman that for a $\mathbf{GL}$-invariant ideal $I \subseteq S$, the linear strands of its minimal free resolution translates via the BGG correspondence to modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. When $I=I_λ$ is the ideal generated by the $\mathbf{GL}$-orbit of a highest weight vector of weight $…
Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in terms of the previous t_i. As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t_i. We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t_1.
Hierarchical structure of graded Betti numbers in the quadratic strand
本文研究graded Betti数的分层结构，建立了一般性的层次体系，并证明了广义K_{p,1}定理，揭示了二次链中极值行为的完整描述。
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(\bf b)$ for a degree $\mathbf{b}\in \A$. We show that the homology of $D_{gcd}(\bf b)$ determines the $i$-Betti numbers of degree $\bf b$. We discuss the notion of an indispensable c…
We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for…
Herzog and Srinivasan have conjectured that for any homogeneous k-algebra, the degree is bounded above by a function of the maximal degrees of the syzygies. Combining the syzygy quadrangle decomposition of Peeva and Sturmfels and a delicate case analysis, we prove that this conjectured bound holds for codimension 2 lattice ideals.
Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g_1,...,g_n of degree d in B=k[x,y] which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix phi for the row vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals of phi to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterized planar curve C which c…
Harm Derksen made a conjecture concerning degree bounds for the syzygies of rings of polynomial invariants in the non-modular case. We provide counterexamples to this conjecture, but also prove a slightly weakened version. We also prove some general results that give degree bounds on the homology of complexes and of Tor groups.
For an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0 --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby present the Artin-Rees number as a proper replacement of regularity in the local situation, --> the ring R/I^k is Golod, its Poincer{é}-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of work of Kodiyalam and Cut…
Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules
本文研究完全交环上极大 Cohen-Macaulay 模的高阶合冲，证明其 Hilbert 系数呈周期 2 的拟多项式渐近规律，并在复杂度为 2 且等号成立时给出相伴分次环正则性的一致有界性。
Quadratic equations of tangent varieties via four-way tensors of linear forms
该研究利用四向线性形式张量构造切簇的二次方程与线性合冲并推广至高阶情形，拓展了经典行列式方程，证明了相关猜想变体，还从表示论角度诠释了Green-Lazarsfeld类。
If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=π_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a module over the exterior algebra E on \A: θ_k(G) = \dim_\k Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, θ_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\A). Our earlier wor…
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded m…
$k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions
本文通过杠分解和 dg 或 $A_\infty$ 分解，对正 Burch 指数环上的模合冲中剩余域直和项的存在性给出新证明，并明确识别 Golod 模情形下的 $k$ 个直和项，其数量随同调度指数增长。
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
Cuspidal plane curves, syzygies and a bound on the MW-rank
本文研究度为6k且仅含节点和普通尖点的约化平面曲线，通过证明其尖点理想极小消解中b_i≤5k，给出椭圆三维流形Mordell-Weil秩的精确刻画与上界，并将Alexander多项式中(t²-t+1)指数的上界较Cogolludo-Libgober结果改进近一倍。
The fundamental representations of the special linear group ${\rm SL}_n$ over the complex numbers are the exterior powers of $\mathbb{C}^n$. We consider the invariant rings of sums of arbitrary many copies of these ${\rm SL}_n$-modules. The symbolic method for antisymmetric tensors developed by Grosshans, Rota and Stein is used, but instead of brackets, we associate colored hypergraphs to the invariants. This approach allows us to use results and insights from graph theory. In particular, we determine (minimal) generating sets of the invariant rings in the case of ${\rm SL}_4$ and ${\rm SL}_5$…
We establish doubly-exponential degree bounds for Gröbner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dubé based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the c…
We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies with the standard method provided by the Extended Euclidean Algorithm. As a consequence, we obtain explicit descriptions for solutions of "minimal" degrees in terms of the degrees of elements appearing in the EEA. This allows us to describe the minimal degree in a $μ$-basis of a polynomial planar parametrization in terms of a "critical" degree arising in the EEA.
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