The initial ideal of generic sequences and Fr\"{o}berg's Conjecture

TL;DR

This paper investigates Pardue's Conjecture regarding the initial ideal of generic sequences and its equivalence to Fr"{o}berg's Conjecture, providing partial proofs under certain degree conditions and advancing understanding of Hilbert functions of generic ideals.

Contribution

The paper proves Pardue's Conjecture under specific degree conditions, offering a partial resolution to Fr"{o}berg's Conjecture for cases where r ≤ n+2.

Findings

Pardue's Conjecture holds under certain degree conditions.

Partial proof of Fr"{o}berg's Conjecture for r ≤ n+2.

Results apply over any characteristic field.

Abstract

Let $K$ be an infinite field and let $I = (f_1,\cdots,f_r)$ be an ideal in the polynomial ring $R = K[x_1,\cdots,x_n]$ generated by generic forms of degrees $d_1,\cdots,d_r$. A longstanding conjecture by Fr\"{o}berg predicts the shape of the Hilbert function of $R/I.$ In 2010 Pardue stated a conjecture on the initial ideal of $n$ generic forms with respect to the deg-revlex order and he proved that it is equivalent to Fr\"{o}berg's Conjecture. We study Pardue's Conjecture and we prove it under suitable conditions on the degrees of the forms. This yields a partial solution to Fr\"{o}berg's Conjecture in the case $r \leq n+2$ over an infinite field of any characteristic.

Equations

$$H{S}_A(z)=\frac{{\prod }_{i=1}^n(1-z^{d_i})}{(1-z{)}^n}.$$

$$A_t\stackrel{f}{\to }{A}_{t+d}$$

$$H{F}_{A/fA}(t)=\max \{0,H{F}_A(t)-H{F}_A(t-d)\}.$$

$$H{S}_{A/fA}(z)=\lceil (1-z^j)H{S}_A(z)\rceil ,$$

$$H{S}_A(z)=\lceil \frac{{\prod }_{i=1}^r(1-z^{d_i})}{(1-z{)}^n}\rceil .$$

$$d_i\ge \min \left\{d_1+\cdots +d_{i-2}-i+2,\lfloor \frac{d_1+\cdots +d_{i-1}-i+1}{2}\rfloor \right\}$$

$$0<a_0<a_1<\cdots <a_{\sigma }=\cdots =a_{\delta -\sigma }>\cdots >a_{\delta -1}>a_{\delta }>0.$$

$${\tilde{B}}_i^0=\{x^{\mu }\in B_i|\max (x^{\mu })<n\}.$$

$$B_0=\{1\}$$

$$B_1={\tilde{B}}_1^0\cup \{x_4\}\text{where}{\tilde{B}}_1^0=\{x_1,x_2,x_3\}$$

$$B_2={\tilde{B}}_2^0\cup x_4{B}_1\text{where}{\tilde{B}}_2^0=\{x_1{x}_2,x_2^2,x_1{x}_3,x_2{x}_3,x_3^2\}$$

$$B_3={\tilde{B}}_3^0\cup x_4{B}_2\text{where}{\tilde{B}}_3^0=\{x_1{x}_2{x}_3,x_2^2{x}_3,x_1{x}_3^2,x_2{x}_3^2,x_3^3\}$$

$$B_4={\tilde{B}}_4^0\cup x_4{B}_3\text{where}{\tilde{B}}_4^0=\{x_2{x}_3^3,x_3^4\}$$

$$B_5=x_4^2{B}_3$$

$$B_6=x_4^4{B}_2$$

$$B_7=x_4^6{B}_1$$

$$B_8=x_4^8{B}_0$$

$$E_i=B_i\cup z{B}_{i-1}\cup z^2{B}_{i-2}\cup \cdots \cup z^{i-1}{B}_1\cup z^i{B}_0,$$

$${\mathbf{E}}_i.g\equiv M_i{\mathbf{E}}_{i+d}\text{(mod}G),$$

$$\mathrm{in}(I,g)=(\mathrm{in}(I),z^{d-\delta }{B}_{\delta },z^{d-\delta +2}{B}_{\delta -1},\cdots ,z^{\delta +d-2}{B}_1,z^{\delta +d}{B}_0).$$

$$\mathrm{in}(I,g)=(\mathrm{in}(I),x_n^{\delta },z^2{B}_{\delta -1},\cdots ,z^{2\delta -2}{B}_1,z^{2\delta }{B}_0).$$

$$g={\mathbf{v}}_d{\mathbf{B}}_d+{\mathbf{v}}_{d-1}{\mathbf{B}}_{d-1}z+\cdots +{\mathbf{v}}_1{\mathbf{B}}_1{z}^{d-1}+{\mathbf{v}}_0{z}^d,$$

$$\begin{array}{cc}& \begin{array}{ccccccc}{\mathbf{B}}_{d+i}& z{\mathbf{B}}_{d+i-1}& \cdots & z^{i-1}{\mathbf{B}}_{d+1}& z^i{\mathbf{B}}_d& \cdots & z^{d+i}\end{array}\\ \left(\begin{array}{c}{\mathbf{B}}_i\\ z{\mathbf{B}}_{i-1}\\ ⋮\\ z^{i-1}{\mathbf{B}}_1\\ z^i{\mathbf{B}}_0\end{array}\right).g\equiv & \left(\begin{array}{ccccccc}{\Gamma }_{i,d+i}& {\Gamma }_{i,d+i-1}& \cdots & {\Gamma }_{i,d+1}& {\Gamma }_{i,d}& \cdots & {\Gamma }_{i,0}\\ 0& {\Gamma }_{i-1,d+i-1}& \cdots & {\Gamma }_{i-1,d+1}& {\Gamma }_{i-1,d}& \cdots & {\Gamma }_{i-1,0}\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\Gamma }_{1,d+1}& {\Gamma }_{1,d}& \cdots & {\Gamma }_{1,0}\\ 0& 0& \cdots & 0& {\Gamma }_{0,d}& \cdots & {\Gamma }_{0,0}\end{array}\right)\end{array}$$

$${\mathbf{B}}_i={\tilde{\mathbf{B}}}_i^0\cup x_n{\tilde{\mathbf{B}}}_{i-1}^0\cup \cdots \cup x_n^{i-1}{\tilde{\mathbf{B}}}_1^0\cup \{x_n^i\}.$$

$$x_n^d{\mathbf{B}}_i=x_n^d{\tilde{\mathbf{B}}}_i^0\cup x_n^{d+1}{\tilde{\mathbf{B}}}_{i-1}^0\cup \cdots \cup x_n^{d+i-1}{\tilde{\mathbf{B}}}_1^0\cup \{x_n^{d+i}\}.$$

$$x^{\alpha }{x}_n^d=x_n^{d+j}{x}^{\beta }\in x_n^{d+j}{\tilde{\mathbf{B}}}_{i-j}^0\subset {\mathbf{B}}_{d+i}.$$

$${\Gamma }_{i,d+i}=\left(\begin{array}{cccccc}{L}_{1,1}& \cdots & c_d+L_{1,k}& L_{1,k+1}& \cdots & L_{1,a_{d+i}}\\ L_{2,1}& \cdots & L_{2,k}& c_d+L_{2,k+1}& \cdots & L_{2,a_{d+i}}\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ L_{a_i,1}& \cdots & L_{a_i,k}& L_{a_i,k+1}& \cdots & c_d+L_{a_i,a_{d+i}}\end{array}\right),$$

$$\mathrm{rank}\left(\begin{array}{cc}{\Gamma }_{i,d+i}& {\Gamma }_{i,d+i-1}\\ 0& {\Gamma }_{i-1,d+i-1}\end{array}\right)=a_i+a_{i-1}.$$

$$\begin{array}{cc}& \begin{array}{cccc}{\mathbf{B}}_{d+i}& z{\mathbf{B}}_{d+i-1}& \cdots & z^{d+2i-\delta }{\mathbf{B}}_{\delta -i}\end{array}\\ {\Lambda }_i=& \left(\begin{array}{cccc}{\Gamma }_{i,d+i}& {\Gamma }_{i,d+i-1}& \cdots & {\Gamma }_{i,\delta -i}\\ 0& {\Gamma }_{i-1,d+i-1}& \cdots & {\Gamma }_{i-1,\delta -i}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\Gamma }_{\delta -d-i,\delta -i}\end{array}\right)\end{array}.$$

$$M_i=\left(\begin{array}{cc}{\Lambda }_i& \Omega \\ 0& M_{\delta -d-i-1}\end{array}\right).$$

$$F_0$$

$$F_1$$

$$⋮$$

$$F_{d-1}$$

$$F_d$$

$$F_{d+1}$$

$$⋮$$

Full text

1 Introduction

Let $R=K[x_{1},\cdots,x_{n}]$ be the polynomial ring in $n$ variables over an infinite field $K.$ A homogeneous ideal $I$ in $R$ is said to be of type $(n;d_{1},\cdots,d_{r})$ if it is generated by $r$ forms of degree $d_{i}$ for $i=1,\cdots,r.$ The Hilbert function of $A=R/I$ is by definition $HF_{A}(t):=\operatorname{dim}_{K}R_{t}/I_{t}$ for every $t\geq 0$ and it reflects important information of the ideal $I$. We are interested in the behavior of the Hilbert function of generic ideals of type $(n;d_{1},\cdots,d_{r}).$ We adopt the definition of generic ideals given by Fröberg in [8] because it is more suitable for our approach. Assume that $K$ is an extension of a base field $F$.

Definition 1.1**.**

(1) A form of degree $d$ in $K[x_{1},\cdots,x_{n}]$ is called generic over $F$ if it is a linear combination of all monomials of degree $d$ and all coefficients are algebraically independent over $F$.**

(2) A homogeneous ideal $(f_{1},\cdots,f_{r})$ is called generic if all $f_{i}$ are generic forms with all the coefficients algebraically independent over $F$.**

Other definitions in the literature appear in terms of the affine space whose points are the coefficients of the $r$ forms and we say that a property of such sequences is generic if it holds in a nonempty Zariski open subset of the affine space, see for instance [16] or [17]. The property ought to hold for a randomly chosen sequence. The Hilbert function of a generic ideal of type $(n;d_{1},\cdots,d_{n})$ is the Hilbert function of a regular sequence, hence the generating Hilbert series $HS_{A}(z):=\sum_{t\geq 0}HF_{A}(t)z^{t}$ is well known

[TABLE]

In general if $A$ is a graded standard $K$-algebra and $f\in R_{d}$ is a generic form, it is natural to guess that for every $t\geq 0,$ the multiplication map

[TABLE]

is of maximal rank (either injective or surjective), hence

[TABLE]

In terms of the corresponding Hilbert series this can be rewritten as

[TABLE]

where for a power series $\sum a_{i}z^{i}$ one has $\lceil\sum a_{i}z^{i}\rceil=\sum b_{i}z^{i}$, with $b_{i}=a_{i}$ if $a_{j}>0$ for all $j\leq i$, and $b_{i}=0$ otherwise. Following the terminology of Pardue in [17], if this is the case, we say that $f$ is semi-regular. A sequence of homogeneous polynomials $f_{1},\cdots,f_{r}$ is a semi-regular sequence on $A$ if each $f_{i}$ is semi-regular on $A/(f_{1},\cdots,f_{i-1})$. Clearly regular sequences are semi-regular sequences. This approach motivated a longstanding conjecture stated in 1985 by Fröberg.

Conjecture 1.2**.**

(Fröberg’s Conjecture). Let $I=(f_{1},\cdots,f_{r})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{r})$ in $R=K[x_{1},\cdots,x_{n}]$. Then the Hilbert series of $A=R/I$ is given by

[TABLE]

This problem is of central interest in commutative algebra in the last decades and a great deal was done (see for instance Anick [2], Fröberg [8], Fröberg-Hollman [9], Fröberg-Löfwall [10], Fröberg-Lundqvist [11], Moreno-Socías [16], Pardue [17], Stanley [19], Valla [20]). A large number of validations through computational methods push to a positive answer. Fröberg’s Conjecture is known to be true for $r\leq n$ (complete intersection); $n=2$ [8]; $n=3$ [2]; $r=n+1$ (almost complete intersection) with char$K=0$ [8]; $d_{1}=\cdots=d_{r}=2$ and $n\leq 11$, $d_{1}=\cdots d_{r}=3$ and $n\leq 8$ [9].

Denote by $\operatorname{in}_{\tau}(I)$ the initial ideal of $I$ with respect to a term order $\tau$ on $R$. Because the Hilbert series of $R/I$ and of $R/\operatorname{in}_{\tau}(I)$ coincide for every $\tau$, a rich literature has been developed with the aim to characterize the initial ideal of generic ideals with respect to suitable term orders. In the case $r=n,$ Moreno-Socías stated a conjecture describing the initial ideal of generic ideals with respect to the degree reverse lexicographic order. From now on, the initial ideal of $I$ will be always with respect to the degree reverse lexicographic order and it will be denoted simply by $\operatorname{in}(I).$ For general facts and properties on the degree reverse lexicographic order see for instance [7, Proposition 15.2]. It is natural to guess generic complete intersections share special initial ideals. We present here Moreno-Socías’ Conjecture.

Conjecture 1.3**.**

(See Moreno-Socías [15].) Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$. Then $\operatorname{in}(I)$ is almost reverse lexicographic, i.e, if $x^{\mu}$ is a minimal generator of $\operatorname{in}(I)$ then every monomial of the same degree and greater than $x^{\mu}$ must be in $\operatorname{in}(I)$ as well.

Moreno-Socías’ Conjecture was proven in the case $n=2$ by Aguire et al. [1] and Moreno-Socías [16], $n=3$ by Cimpoeas [6], $n=4$ by Harima and Wachi [14], and sequences $d_{1},\cdots,d_{n}$ satisfying $d_{i}>\sum_{j=1}^{i-1}d_{j}-i+1$ for every $i\geq 4$ by Cho and Park assuming char$K=0$ [5]. Without restriction on the characteristic of $K,$ by using an incremental method from [12], Capaverde and Gao improved the result of Cho and Park. They gave a complete description of the initial ideal of $I$ in the case $d_{1},\cdots,d_{n}$ satisfies $d_{i}\geq\sum_{j=1}^{i-1}d_{j}-i-1$ for every $i$. In particular they proved Moreno-Socías’ Conjecture under the above conditions, see [4, Theorem 3.19]. Moreno-Socías’ Conjecture implies Fröberg’s Conjecture and Pardue stated a conjecture which is equivalent to Fröberg’s Conjecture ([17, Theorem 2]) by a slight modification of the requirement on the initial ideal.

For every monomial $x^{\alpha}\in K[x_{1},\cdots,x_{n}]$, denote by $\max(x^{\alpha})$ the largest index $i$ such that $x_{i}$ divides $x^{\alpha}$. It is easy to see that Moreno-Socías’ Conjecture implies the following conjecture stated by Pardue.

Conjecture 1.4**.**

[17, Conjecture E]** Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$. If $x^{\mu}$ is a minimal generator of $\operatorname{in}(I)$ and $\max(x^{\mu})=m$ then every monomial of the same degree in the variables $x_{1},\cdots,x_{m-1}$ must be in $\operatorname{in}(I)$ as well.

Our aim is to study Fröberg’s Conjecture directly through the above conjecture instead of Moreno-Socías’ Conjecture.

Definition 1.5**.**

Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal in $K[x_{1},\cdots,x_{n}]$. Suppose $x^{\alpha}$ is a generator of $\operatorname{in}(I)$ and $\max(x^{\alpha})=m$. We say $x^{\alpha}$ satisfies property $P$ if every monomial of the same degree in the variables $x_{1},\cdots,x_{m-1}$ is also in $\operatorname{in}(I)$.**

Thus, in order to claim Conjecture 1.4 is true for $I$, we need to prove that every minimal generator of $\operatorname{in}(I)$ satisfies property $P$. By using the incremental method from [4] and [12], in Proposition 3.3, we give an explicit description of the initial ideal of generic ideals with respect to the degree reverse lexicographic order. This is a crucial point for proving Theorem 3.1 that gives an inductive method for proving Conjecture 1.4.

In Theorem 4.3, we prove that Conjecture 1.4 holds true for every generic ideal if $n\leq 3$. If $n\geq 4$, in Theorem 4.4, we prove Conjecture 1.4 for generic ideals of type $(n;d_{1},\cdots,d_{n})$ where $d_{1}\leq\cdots\leq d_{n}$ and

[TABLE]

for every $4\leq i\leq n.$ Here $\left\lfloor a\right\rfloor$ denotes the greatest integer less or equal to $a.$

As application, in Theorem 4.6, we prove that Fröberg’s Conjecture holds true for generic ideals of type $(n;d_{1},\cdots,d_{r})$ with $r\leq 3$ or $r\geq 4$ and $d_{1}\leq\cdots\leq d_{r}$ verifying condition (1). Our hope is that our results will be successfully applied to give new insights in proving Fröberg’s Conjecture. All the computations in this paper have been performed by using Macaulay2 [13].

We would like to point out that, at the end of our paper, we realized that independently Bertone and Cioffi in [3, Corollary 3.9] proved Moreno-Socías’ Conjecture with a condition similar to (1) under the assumption char$K=0.$ The methods and the aim are lightly different even if strictly related.

2 Preliminaries

Let $R^{\prime}=K[x_{1},\cdots,x_{n},z]$ be the polynomial ring in $n+1$ variables and fix the order on the variables $x_{1}>\cdots>x_{n}>z$. Let $(I,g)=(f_{1},\cdots,f_{n},g)$ be a generic homogeneous ideal of type $(n+1;d_{1},\cdots,d_{n},d)$ in $R^{\prime}$. Note that, $I=(f_{1},\cdots,f_{n})$ is a generic homogeneous ideal of type $(n+1;d_{1},\cdots,d_{n})$ in $R^{\prime}$. Define $\pi:R^{\prime}\longrightarrow R=K[x_{1},\cdots,x_{n}]$ to be the ring homomorphism that takes z to zero, fixing the elements in $K$ and the variables $x_{1},\cdots,x_{n}$. Let $J=\pi(I)$ be the image of $I$. Then $J$ is a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $R$. We recall that we consider always the degree reverse lexicographic order.

Proposition 2.1**.**

$\operatorname{in}(I)$* and  $\operatorname{in}(J)$ have the same minimal generators.*

Proof.

From a property of the degree reverse lexicographic order in [7, Proposition 15.12], we get $\pi(\operatorname{in}(I))=\operatorname{in}(J)$. On the other hand, since $z$ is regular in $R^{\prime}/I$, by [7, Theorem 15.13] $z$ is regular in $R^{\prime}/\operatorname{in}(I)$. Furthermore, by [7, Theorem 15.14] the minimal generators of $\operatorname{in}(I)$ are not divisible by $z$. Thus, $\operatorname{in}(I)$ and  $\operatorname{in}(J)$ have the same minimal generators. ∎

Let $B=B(J)$ be the set of monomials in $R$ which are not in $\operatorname{in}(J)$ and we call them the standard monomials with respect to $J$. We set                                               $\delta=d_{1}+\cdots+d_{n}-n,$                                               $\delta^{*}=d_{1}+\cdots+d_{n-1}-(n-1),$                                               $\sigma=\min\left\{\delta^{*},\left\lfloor\dfrac{\delta}{2}\right\rfloor\right\}.$ It is known that $A=R/J$ is a complete intersection and the Hilbert series of $A$ is a symmetric polynomial of degree $\delta$, say $HS_{A}(z)=\sum_{i=0}^{\delta}a_{i}z^{i},$ with

[TABLE]

Notice that $a_{i}=|B_{i}|$ where $B_{i}$ is the set of monomials of degree $i$ in $B$ (see for instance [16, Proposition 2.2]). The set of standard monomials with respect to a generic ideal of type $(n;d_{1},\cdots,d_{r})$ depends only on $(n;d_{1},\cdots,d_{r})$. So, we denote it by $B(n;d_{1},\cdots,d_{r})$. We will describe more clearly the set of standard monomials in the case $r=n$, that is $B=B(n;d_{1},\cdots,d_{n})$. For each $1\leq i\leq\sigma$, define

[TABLE]

Proposition 2.2**.**

The structure of $B=B(n;d_{1},\cdots,d_{n})$ is as follows, (1) $B_{i}=\widetilde{B}_{i}^{0}\cup x_{n}\widetilde{B}_{i-1}^{0}\cup\cdots\cup x_{n}^{i-1}\widetilde{B}_{1}^{0}\cup\{x_{n}^{i}\}$           $=\widetilde{B}_{i}^{0}\cup x_{n}B_{i-1},$ for all $1\leq i\leq\sigma$. (2) $B_{\sigma+i}=x_{n}^{i}\widetilde{B}_{\sigma}^{0}\cup x_{n}^{i+1}\widetilde{B}_{\sigma-1}^{0}\cup\cdots\cup x_{n}^{\sigma+i-1}\widetilde{B}_{1}^{0}\cup\{x_{n}^{\sigma+i}\}$               $=x_{n}^{i}B_{\sigma}$, for all $0\leq i\leq\delta-2\sigma$. (3) $B_{\delta-i}=x_{n}^{\delta-2i}\widetilde{B}_{i}^{0}\cup x_{n}^{\delta-2i+1}\widetilde{B}_{i-1}^{0}\cup\cdots\cup x_{n}^{\delta-i-1}\widetilde{B}_{1}^{0}\cup\{x_{n}^{\delta-i}\}$               $=x_{n}^{\delta-2i}B_{i}$, for all $0\leq i\leq\sigma$.

Proof.

(1) For $1\leq i\leq\sigma$, we have $a_{i-1}<a_{i}$. Let $S$ denote the subset of $B_{i}$ consisting of the $a_{i-1}$ smallest monomials in $B_{i}$. By [4, Lemma 3.5] we get $S=x_{n}B_{i-1}$. It is clear that $\widetilde{B}_{i}^{0}\subseteq B_{i}\setminus S$. Conversely, for every monomial $x^{\alpha}\in B_{i}\setminus S$. Assume that $x^{\alpha}\notin\widetilde{B}_{i}^{0}$. Then $x_{n}$ divides $x^{\alpha}$, so that $x^{\alpha}/x_{n}\in B_{i-1}$. This implies a contradiction since $x_{\alpha}=x_{n}(x^{\alpha}/x_{n})\in S$. Thus, $\widetilde{B}_{i}^{0}=B_{i}\setminus S$ and (1) holds. (2) For $0\leq i\leq\delta-2\sigma$, we have $a_{\sigma+i}=a_{\sigma}$. By [4, Lemma 3.5] we get $B_{\sigma+i}=x_{n}^{i}B_{\sigma}$. (3) Since $\sigma\leq\dfrac{\delta}{2}$, by [4, Lemma 3.4] we get $B_{\delta-i}=x_{n}^{\delta-2i}B_{i}$ for $0\leq i\leq\sigma$. ∎

Remark 2.3**.**

(1) $B=B(n;d_{1},\cdots,d_{n})$ is determined by $\widetilde{B}_{1}^{0},\cdots,\widetilde{B}_{\sigma}^{0}$. (2) $|\widetilde{B}_{i}^{0}|=a_{i}-a_{i-1}=a^{\prime}_{i}>0$, for all $1\leq i\leq\sigma$.**

In the following example, we show explicitly the structure of $B(n;d_{1},\cdots,d_{n})$ according to Proposition 2.2.

Example 2.4**.**

Let $B=B(4;2,3,3,4)$ be the set of standard monomials with respect to a generic ideal of type $(4;2,3,3,4)$. Then $\delta=8;\sigma=4$. Denote by $a_{i}=|B_{i}|$ and $a^{\prime}_{i}=|\widetilde{B}_{i}^{0}|$. We have

[TABLE]

In the next section, we will see how to construct $B(n+1,d_{1},\cdots,d_{n+1})$ starting from $B(n,d_{1},\cdots,d_{n})$ by using the incremental method introduced in [12] and adapted to our situation in [4].

3 Main results

Let $(I,g)=(f_{1},\cdots,f_{n},g)$ be a generic homogeneous ideal of type $(n+1;d_{1},\cdots,d_{n},d)$ in $R^{\prime}=K[x_{1},\cdots,x_{n},z]$. Define $C_{I}$ to be the set of the coefficients of the polynomials $f_{1},\cdots,f_{n}$ and $\bar{F}=F(C_{I})\subset K$. Let $G=\{g_{1},\cdots,g_{t}\}$ be the reduced Gröbner basis of $I$ with respect to the degree reverse lexicographic order. Then $g_{1},\cdots,g_{t}$ are homogeneous polynomials in $\bar{F}[x_{1},\cdots,x_{n},z]$. Reducing $g$ modulo $G$ we obtain a polynomial that is a $K$-linear combination of all monomials of degree $d$ in $E$ with coefficients that are still algebraically independent over $\bar{F}$. Hence, from now on we assume that $g$ is reduced modulo $G$ and the coefficients of $g$ are algebraically independent over $\bar{F}$. In order to construct $B(n+1,d_{1},\cdots,d_{n+1})$ from $B(n,d_{1},\cdots,d_{n})$ we need to compare $\operatorname{in}(I,g)$ and $\operatorname{in}(I)$. We recall here the incremental method to construct $\operatorname{in}(I,g)$ from $\operatorname{in}(I)$, for more details see [4, Section 3]. Let $E=B(n+1,d_{1},\cdots,d_{n})$ be the set of standard monomials with respect to $I$ and $B=B(n;d_{1},\cdots,d_{n})$ be the set of standard monomials with respect to $J=\pi(I)$. Denote by $E_{i}$ the set of monomials of degree $i$ in $E$ for every $i\geq 0$. Note that, for $0\leq i\leq\delta$, we have

[TABLE]

and for $i>\delta$, we have $E_{i}=z^{i-\delta}E_{\delta}.$

Let $0\leq i\leq\delta$, denote by $\mathbf{E}_{i}$ the column vector whose entries are the monomials in $E_{i}$ listed in decreasing order. For each monomial $x^{\alpha}\in\mathbf{E}_{i}$, reducing the product $x^{\alpha}g\in R^{\prime}_{i+d}$ modulo $G$ we obtain a polynomial, say the reduced form of $x^{\alpha}g$, that is a $K$-linear combination of monomials in $E_{i+d}$. Note that each coefficient of the reduced form of $x^{\alpha}g$ is a $\bar{F}$-linear combination of coefficients of polynomial $g$. Let $M_{i}$ denote the matrix such that

[TABLE]

where $\mathbf{E}_{i+d}$ denotes the column vector whose entries are the monomials in $E_{i+d}$ listed in decreasing order. By [4, Lemma 3.2] the rows of $M_{i}$ are linearly independent. This means that $\operatorname{rank}(M_{i})=|E_{i}|=a_{i}+a_{i-1}+\cdots+a_{0}$. Furthermore, the monomials in $\mathbf{E}_{i+d}$ corresponding to the $|E_{i}|$ first linearly independent columns of $M_{i}$ are the generators that will be added to $\operatorname{in}(I)$ to form $\operatorname{in}(I,g)$. Note that some of the monomials we add might be redundant. In this section, we will prove the following main result.

Theorem 3.1**.**

Let $(I,g)=(f_{1},\cdots,f_{n},g)$ be a generic homogeneous ideal of type $(n+1;d_{1},\cdots,d_{n},d)$ in $R^{\prime}=K[x_{1},\cdots,x_{n},z]$, where $d_{1}\leq\cdots\leq d_{n}\leq d$. If $d\geq\sigma$ and Conjecture 1.4 is true for $J=\pi(I)$, then Conjecture 1.4 is also true for $(I,g)$.

The proof needs a deep investigation given in Proposition 3.3, Proposition 3.6 and Proposition 3.7. First of all we notice that Theorem 3.1 can be deduced from [4, Proposition 3.12] when $d\geq\delta$. Indeed, by [4, Proposition 3.12], if $d\geq\delta$ then

[TABLE]

Let $x^{\mu}$ be a generator of $\operatorname{in}(I,g)$ in $z^{d-\delta}B_{\delta},z^{d-\delta+2}B_{\delta-1},\cdots,z^{\delta+d-2}B_{1},z^{\delta+d}B_{0}$. We claim $x^{\mu}$ satisfies property $P$. Indeed, if $d>\delta$ then $x^{\mu}$ is divisible by $z$ and $\operatorname{deg}(x^{\mu})=k>\delta$. Hence, every monomial $x^{\alpha}$ of degree $k$ in variables $x_{1},\cdots,x_{n}$ is not in $E_{k}=z^{k-\delta}E_{\delta}$, so that $x^{\alpha}\in\operatorname{in}(I)\subset\operatorname{in}(I,g)$. If $d=\delta$ then

[TABLE]

If $x^{\mu}$ is a monomial in $z^{2}B_{\delta-1},\cdots,z^{2\delta-2}B_{1},z^{2\delta}B_{0}$, then $x^{\mu}$ satisfies property $P$ through a similar argument as above. On the other hand, it is not hard to see that $x_{n}^{\delta}$ also satisfies properties P. Thus, if Conjecture 1.4 is true for $J$ with $d\geq\delta$, then every minimal generator of $\operatorname{in}(I,g)$ satisfies property $P$. This means Conjecture 1.4 is true for $(I,g)$. Consider now the case $d<\delta$. Set $i^{*}=\lfloor\frac{\delta-d}{2}\rfloor$. The following lemma will be useful for proving Proposition 3.3.

Lemma 3.2**.**

For $i>j\geq i^{*}$, we have $a_{d+i}\leq a_{d+j}$. Furthermore, the monomials of $B_{d+i}$ are multiples of the $a_{d+i}$ smallest monomials in $B_{d+j}$.

Proof.

Since $d+i^{*}\geq\lfloor\frac{\delta}{2}\rfloor\geq\sigma$ , so $d+i>d+j\geq\sigma$. Hence, $a_{d+i}\leq a_{d+j}$. Let $S$ denote the subset of $B_{d+j}$ consisting of the $a_{d+i}$ smallest monomials in $B_{d+j}$. By [4, Lemma 3.5 (ii)] we have $B_{d+i}=x_{n}^{i-j}S.$ ∎

By convention, we use the following notation. Let $B$ be a finite subset of monomials in $R=K[x_{1},\cdots,x_{n}]$ and denote by $\mathbf{B}=\{x^{\alpha_{1}},\cdots,x^{\alpha_{k}}\}$ the set of monomials in $B$ listed in decreasing order. Let $S$ be a subset of $[1,k]$ and denote $\mathbf{B}^{S}=\{x^{\alpha_{i}}\in B\ |\ i\in S\}$. The set of generators of $\operatorname{in}(I,g)$ can be described as the following.

Proposition 3.3**.**

Let $(I,g)=(f_{1},\cdots,f_{n},g)$ be a generic ideal of type $(n+1;d_{1},\cdots,d_{n},d)$ in $R^{\prime}=K[x_{1},\cdots,x_{n},z]$, where $d_{1}\leq\cdots\leq d_{n}\leq d$ and $d<\delta$. Let $B=B(n;d_{1},\cdots,d_{n})$ be the set of standard monomials with respect to a generic ideal of type $(n;d_{1},\cdots,d_{n})$ in $R=K[x_{1},\cdots,x_{n}]$. (1) If $\delta-d\equiv 0$ ( $\operatorname{mod}2$) then

$\operatorname{in}(I,g)=(\operatorname{in}(I),\mathbf{B}_{d}^{\{1\}},\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}},z^{2}B_{d+i^{*}-1},\cdots,z^{\delta-d}B_{d},$

$z^{\delta-d+2}B_{d-1},\cdots,z^{\delta+d-2}B_{1},z^{\delta+d}B_{0}),$**

(2) If $\delta-d\equiv 1$ ( $\operatorname{mod}2$) then

$\operatorname{in}(I,g)=(\operatorname{in}(I),\mathbf{B}_{d}^{\{1\}},\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i*}},B_{d+i^{*}+1},zB_{d+i^{*}},\cdots,z^{\delta-d}B_{d},$

$z^{\delta-d+2}B_{d-1},\cdots,z^{\delta+d-2}B_{1},z^{\delta+d}B_{0}),$**

where $S_{i}$ is a subset of $[1,a_{d+i}]$ containing $a_{i}$ elements for every $i=1,\cdots,i^{*}$.

Proof.

Since $g$ is a combination of all monomials in $E_{d}=B_{d}\cup zB_{d-1}\cup\cdots\cup z^{d}B_{0}$, we can write

[TABLE]

where $\mathbf{v}_{i}$ is the row vector of the coefficients of $g$ corresponding to the monomials in $\mathbf{B}_{i}$. Denote the last coefficient of $\mathbf{v_{d}}$ by $c_{d}$. Note that $c_{d}$ is the coefficient corresponding to the monomial $x_{n}^{d}$. Set $\mathbf{v}_{d}^{*}=\mathbf{v}_{d}\setminus\{c_{d}\}$. We will construct a set of generators for $\operatorname{in}(I,g)$ by using incremental method. For $i=0$, we have $E_{0}=\{1\}$ and $M_{0}$ is a row matrix $\begin{pmatrix}\mathbf{v}_{d}&\mathbf{v}_{d-1}&\cdots&\mathbf{v}_{1}&\mathbf{v}_{0}\end{pmatrix}$. Hence, the first column of $M_{0}$ is linearly independent. Thus, the largest monomial of $\mathbf{B}_{d}$ will be added. For $1\leq i\leq i^{*},$ equation (2) can be explicitly written in the following form

[TABLE]

Denote by $A_{i},A_{i-1},\cdots,A_{1},A_{0}$ the submatrices of $M_{i}$ formed by the columns corresponding to the monomials in $\mathbf{B}_{d+i},z\mathbf{B}_{d+i-1},\cdots,z^{i-1}\mathbf{B_{d+1}},z^{i}\mathbf{B_{d}}$ respectively. We now consider the block $\Gamma_{i,d+i}$. Note that the entries of $\Gamma_{i,d+i}$ are the $\bar{F}$-linear combinations of the coefficients in $\mathbf{v}_{d}$. Since $i\leq\dfrac{\delta-d}{2}$, we have $i<\sigma$ and

[TABLE]

Since $d+i\leq\delta-i$, we have $a_{d+i}\geq a_{i}$ and the $a_{i}$ smallest monomials in $\mathbf{B}_{d+i}$ are

[TABLE]

Let $x^{\alpha}\in\mathbf{B}_{i}$. Suppose $x^{\alpha}=x_{n}^{j}x^{\beta}\in x_{n}^{j}\widetilde{\mathbf{B}}_{i-j}^{0}$, for some $0\leq j\leq i$ and $x^{\beta}\in\widetilde{\mathbf{B}}_{i-j}^{0}$ . Then

[TABLE]

Thus, the term $c_{d}.{x^{\alpha}}x_{n}^{d}$ of the product $x^{\alpha}g$ is reduced mod $G$. Therefore, in the coefficients of the reduced form of the product $x^{\alpha}.g$, $c_{d}$ will appear only in the coefficient of the monomial $x_{n}^{d+j}x^{\beta}\in\mathbf{\mathbf{B}}_{d+i}$. Thus,

[TABLE]

where $k=a_{d+i}-a_{i}+1$ and $L_{i,j}$ is a $\bar{F}$-linear combination of the coefficients in $\mathbf{v}_{d}^{*}$. Hence, the $a_{i}$ last columns of $\Gamma_{i,d+i}$ are linearly independent. So, $\operatorname{rank}(\Gamma_{i,d+i})=a_{i}$. This implies that the $a_{i}$ first linearly independent columns of $M_{i}$ are the $a_{i}$ first linearly independent columns of $A_{i}$ (before defined). Define $S_{i}$ to be the subset of $[1,a_{d+i}]$ such that its elements are the indices of the $a_{i}$ first linearly independent columns of $A_{i}$. Then the generators which will be added are the elements in $\mathbf{B}_{d+i}^{S_{i}}$. Since the $a_{i-1}$ last columns of $\Gamma_{i-1,d+i-1}$ are linearly independent again, we have

[TABLE]

Therefore, the $a_{i-1}$ next linearly independent columns of $M_{i}$ are the $a_{i-1}$ first linearly independent columns of $A_{i-1}$ and so on. We have the $a_{i}+a_{i-1}+\cdots+a_{0}$ first linearly independent columns of $M_{i}$ are the $a_{i},a_{i-1},\cdots,a_{1},a_{0}$ first linearly independent columns of $A_{i},A_{i-1},\cdots,A_{1},A_{0}$ respectively.

However, the monomials in $z\mathbf{B}_{d+i-1},\cdots,z^{i-1}\mathbf{B}_{d+1},z^{i}\mathbf{B}_{d}$ corresponding to the first linearly independent columns of $A_{i-1},\cdots,A_{1},A_{0}$ respectively are redundant since they are multiples of the monomials that were added in the steps $i-1,\cdots,1,0$. Thus, in the step $i$, only the monomials $\mathbf{B}_{d+i}^{S_{i}}$ will be added.

Note that, in the case $\delta-d\equiv 0$ (mod $2$) we have $\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}=B_{d+i^{*}}$. Indeed, since $d+i^{*}=\delta-i^{*}$, by [4, Lemma 3.4] we have $a_{d+i^{*}}=a_{\delta-i^{*}}=a_{i^{*}}$. Hence, $S_{i^{*}}=[1,a_{d+i^{*}}]$. For $i^{*}<i<\delta-d$, equation (2) also has form as in (3). Let $\Lambda_{i}$ denote the square submatrix of $M_{i}$ given by

[TABLE]

Then, $M_{i}$ has form

[TABLE]

By [4, Proposition 3.16] $\Lambda_{i}$ is nonsingular. Hence, the first linearly independent columns of $M_{i}$ are given by all the columns of $\Lambda_{i}$ and the columns corresponding to the first linearly independent columns of $M_{\delta-d-i-1}$. Note that the monomials in $E_{d+i}$ corresponding to the first linearly independent columns of $M_{\delta-d-i-1}$ are redundant since they are multiplies of monomials were already added in the step $\delta-d-i-1$. Furthermore, by using Lemma 3.2,for $i=i^{*}+1,\cdots,\delta-d-1$, we obtain the following. $\bullet$ If $\delta-d\equiv 0$ (mod $2$) then only the monomials in $z^{2}B_{d+i^{*}-1},\cdots,z^{\delta-d-2}B_{d+1}$ will be added. $\bullet$ If $\delta-d\equiv 1$ (mod $2$) then only the monomials in $B_{d+i^{*}+1},zB_{d+i^{*}},\cdots,z^{\delta-d-2}B_{d+1}$ will be added. Finally, for $\delta-d\leq i\leq\delta$, by [4, Corollary 3.11] the monomials in $z^{\delta-d}B_{d}$ , $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$ will be added. ∎

Remark 3.4**.**

(1) The set of generators of $\operatorname{in}(I,g)$ appearing in Proposition 3.3 is not minimal. For instance the monomial $z^{\delta-d}\mathbf{B}_{d}^{\{1\}}$ is a multiple of $\mathbf{B}_{d}^{\{1\}}$. However, with respect to the property P, we will see that the problems only come from the monomials in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$. (2) The set of generators of $\operatorname{in}(I,g)$ in Proposition 3.3 is determined by the monomials in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$.**

The following lemma will be useful for proving Proposition 3.6.

Lemma 3.5**.**

$B_{d+i}\subset\operatorname{in}(I,g)$* for every $i>i^{*}$.*

Proof.

If $\delta-d\equiv 0$ (mod $2$), by Proposition 3.3 $B_{d+i^{*}}\subset\operatorname{in}(I,g)$. Hence, by Lemma 3.2 $B_{d+i}\subset\operatorname{in}(I,g)$. If $\delta-d\equiv 1$ (mod $2$) then $B_{d+i^{*}+1}\subset\operatorname{in}(I,g)$, and Lemma 3.5 follows from Lemma 3.2 again. ∎

In the next proposition, we will prove that the generators of $\operatorname{in}(I,g)$ not in $\operatorname{in}(I)$, $\mathbf{B}_{d}^{\{1\}}$, $\mathbf{B}_{d+1}^{S_{1}}$, $\cdots$, $\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$ satisfy property $P$ in any case.

Proposition 3.6**.**

(1) If $\delta-d\equiv 0$ (mod $2$), then the generators of $\operatorname{in}(I,g)$ in $z^{2}B_{d+i^{*}-1}$, $\cdots$, $z^{\delta-d}B_{d}$, $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$ satisfy property $P$. (2) If $\delta-d\equiv 1$ (mod $2$), then the generators of $\operatorname{in}(I,g)$ in $B_{d+i^{*}+1}$, $zB_{d+i^{*}}$, $\cdots$, $z^{\delta-d}B_{d}$, $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$ satisfy property $P$.

Proof.

(1) If $x^{\mu}$ is a generator of $\operatorname{in}(I,g)$ in $z^{2}B_{d+i^{*}-1}$, $\cdots$, $z^{\delta-d}B_{d}$, $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$, then $x^{\mu}$ is divisible by $z$ and $\deg(x^{\mu})=k>d+i^{*}$. For every monomial $x^{\alpha}$ of degree $k$ in variables $x_{1},\cdots,x_{n}$, if $x^{\alpha}\in B_{k}$, by Lemma 3.5, then we have $x^{\alpha}\in\operatorname{in}(I,g)$. Otherwise $x^{\alpha}\notin B_{k}$, so that $x^{\alpha}\in\operatorname{in}(I)\subset\operatorname{in}(I,g)$. Thus, $x^{\mu}$ satisfies property $P$.

(2) If $x^{\mu}$ is a generator of $\operatorname{in}(I,g)$ in $B_{d+i^{*}+1}$, then $x^{\mu}$ is divisible by $x_{n}$ because $B_{d+i^{*}+1}=x_{n}^{d+1}B_{i^{*}}$. Hence, for every monomial $x^{\alpha}$ of degree $d+i^{*}+1$ in variables $x_{1},\cdots,x_{n-1}$, we have $x^{\alpha}\notin B_{d+i^{*}+1}$, so that $x^{\alpha}\in\operatorname{in}(I)\subset\operatorname{in}(I,g)$. Thus, $x^{\mu}$ satisfies property $P$.

If $x^{\mu}$ is a generator of $\operatorname{in}(I,g)$ in $zB_{d+i^{*}}$, $\cdots$, $z^{\delta-d}B_{d}$, $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$. Then, by an argument as in (1), $x^{\mu}$ satisfies property $P$. ∎

We still have to prove that the generators of $\operatorname{in}(I,g)$ in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$ satisfy property $P$. Under condition $d\geq\sigma$, we get the following.

Proposition 3.7**.**

If $d\geq\sigma$, then the generators of $\operatorname{in}(I,g)$ in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$ satisfy property $P$.

Proof.

Since $d\geq\sigma$ , by Propsition 2.2 the monomials in $B_{d+1},\cdots,B_{d+i^{*}}$ are divisible by $x_{n}$. Hence, if $x^{\mu}$ is a generator of $\operatorname{in}(I,g)$ in $\mathbf{B}_{d+i}^{S_{i}}$, for some $1\leq i\leq i^{*}$, then $x^{\mu}$ is divisible by $x_{n}$. This follows $x^{\mu}$ satisfies property $P$ because for every monomial $x^{\alpha}$ of degree $d+i$ in variables $x_{1},\cdots,x_{n-1}$, we have $x^{\alpha}\notin B_{d+i}$, so that $x^{\alpha}\in\operatorname{in}(I)\subset\operatorname{in}(I,g)$. ∎

Proof of Theorem 3.1. If $\delta-d\equiv 0$ (mod $2$), by Proposition 3.3 we have

$\operatorname{in}(I,g)=(\operatorname{in}(I),\mathbf{B}_{d}^{\{1\}},\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}},z^{2}B_{d+i^{*}-1},\cdots,z^{\delta-d}B_{d},$

$z^{\delta-d+2}B_{d-1},\cdots,z^{\delta+d-2}B_{1},z^{\delta+d}B_{0}).$

The monomial $\mathbf{B}_{d}^{\{1\}}$ satisfies property $P$ because it is the largest monomial of $\mathbf{B}_{d}$. By Proposition 3.7 and Proposition 3.6 (1) the generators of $\operatorname{in}(I,g)$ in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$ and in $z^{2}B_{d+i^{*}-1}$, $\cdots$, $z^{\delta-d}B_{d}$, $z^{\delta-d+2}B_{d-1}$, $\cdots$, $z^{\delta+d-2}B_{1}$, $z^{\delta+d}B_{0}$ satisfy property $P$. Hence, if Conjecture 1.4 is true for $J=\pi(I)$, then every minimal generator of $\operatorname{in}(I,g)$ satisfies property $P$, so that Conjecture 1.4 is true for $\operatorname{in}(I,g)$.

If $\delta-d\equiv 1$ (mod $2$), then Theorem 3.1 is proved by a similar argument as above. From Proposition 3.3, we have the following corollary that describes more explicitly the set of the standard monomials with respect to $(I,g)$ in case $d<\delta$.

Corollary 3.8**.**

*Let $(I,g)=(f_{1},\cdots,f_{n},g)$ be a generic homogeneous ideal of type $(n+1;d_{1},\cdots,d_{n},d)$ in $K[x_{1},\cdots,x_{n},z]$, where $d_{1}\leq\cdots\leq d_{n}\leq d$ and $d<\delta$. Let $B=B(n;d_{1},\cdots,d_{n})$ and $F=B(n+1;d_{1},\cdots,d_{n},d)$.

(1) If $\delta-d\equiv 0$ (mod $2$), then

$\displaystyle F_{0}$ $\displaystyle=B_{0}$

$\displaystyle F_{1}$ $\displaystyle=B_{1}\cup zF_{0}$

$\displaystyle\ \vdots$

$\displaystyle F_{d-1}$ $\displaystyle=B_{d-1}\cup zF_{d-2}$

$\displaystyle F_{d}$ $\displaystyle=\mathbf{B}_{d}^{(1,a_{d}]}\cup zF_{d-1}$

$\displaystyle F_{d+1}$ $\displaystyle=\mathbf{B}_{d+1}^{[1,a_{d+1}]\backslash S_{1}}\cup zF_{d}$

$\displaystyle\ \vdots$

$\displaystyle F_{d+i^{*}-1}$ $\displaystyle=\mathbf{B}_{d+i^{*}-1}^{[1,a_{d+i^{*}-1}]\backslash S_{i^{*}-1}}\cup zF_{d+i^{*}-2}$

$\displaystyle F_{d+i^{*}}$ $\displaystyle=zF_{d+i^{*}-1}$

$\displaystyle\ \vdots$

$\displaystyle F_{\delta}$ $\displaystyle=z^{\delta-d+1}F_{d-1}$

$\displaystyle F_{\delta+1}$ $\displaystyle=z^{\delta-d+3}F_{d-2}$

$\displaystyle\ \vdots$

$\displaystyle F_{\delta+d-1}$ $\displaystyle=z^{\delta+d-1}F_{0}.$

(2) If $\delta-d\equiv 1$ (mod $2$), then

$\displaystyle F_{0}$ $\displaystyle=B_{0}$

$\displaystyle F_{1}$ $\displaystyle=B_{1}\cup zF_{0}$

$\displaystyle\ \vdots$

$\displaystyle F_{d-1}$ $\displaystyle=B_{d-1}\cup zF_{d-2}$

$\displaystyle F_{d}$ $\displaystyle=\mathbf{B}_{d}^{(1,a_{d}]}\cup zF_{d-1}$

$\displaystyle F_{d+1}$ $\displaystyle=\mathbf{B}_{d+1}^{[1,a_{d+1}]\backslash S_{1}}\cup zF_{d}$

$\displaystyle\ \vdots$

$\displaystyle F_{d+i^{*}}$ $\displaystyle=\mathbf{B}_{d+i^{*}}^{[1,a_{d+i^{*}}]\backslash S_{i^{*}}}\cup zF_{d+i^{*}-1}$

$\displaystyle F_{d+i^{*}+1}$ $\displaystyle=z^{2}F_{d+i^{*}-1}$

$\displaystyle\ \vdots$

$\displaystyle F_{\delta}$ $\displaystyle=z^{\delta-d+1}F_{d-1}$

$\displaystyle F_{\delta+1}$ $\displaystyle=z^{\delta-d+3}F_{d-2}$

$\displaystyle\ \vdots$

$\displaystyle F_{\delta+d-1}$ $\displaystyle=z^{\delta+d-1}F_{0}.$

Thus, in order to construct $F=B(n+1,d_{1},\cdots,d_{n},d)$ from $B=B(n;d_{1},\cdots,d_{n})$ we only need to know explicitly the monomials in $\mathbf{B}_{d+1}^{S_{1}},\cdots,\mathbf{B}_{d+i^{*}}^{S_{i^{*}}}$. In the following example, we help the reader to construct $\operatorname{in}(I,g)$ from $\operatorname{in}(I)$ and $B(n;d_{1},\cdots,d_{n})$ according to Proposition 3.3. Moreover, we construct $F=B(n+1;d_{1},\cdots,d_{n},d)$ from $B(n;d_{1},\cdots,d_{n})$ according to Corollary 3.8.

Example 3.9**.**

Let $(I,g)=(f_{1},\cdots,f_{4},g)$ be the generic ideal of type $(5;2,3,3,4,5)$ in $K[x_{1},\cdots,x_{4},z]$. Let $B=B(4;2,3,3,4)$ as in Example 2.4. Then $\delta=8,\sigma=4,d=5$ and $i^{*}=\lfloor\frac{\delta-d}{2}\rfloor=1$. We write $g$ as reduced form as the following**

[TABLE]

For $i=0$, The largest monomials of $\mathbf{B}_{5}$ will be added, in this case it is $x_{1}x_{2}x_{3}^{2}x_{4}^{2}$.**

For $i=1,$

[TABLE]

where $\mathbf{B}_{1}=\mathbf{\widetilde{B}}_{1}^{0}\cup\{x_{4}\}$ and $\mathbf{B}_{6}=x_{4}^{4}\mathbf{\widetilde{B}}_{2}^{0}\cup x_{4}^{5}\mathbf{\widetilde{B}}_{1}^{0}\cup\{x_{4}^{6}\}$. The monomials in $\mathbf{B}_{6}$ corresponding to the $a_{1}=4$ first linearly independent columns of $\Gamma_{1,6}$ will be added. By using Macaulay2 to compute $\operatorname{in}(I,g)$, we see that the $4$ largest monomials of $\mathbf{B}_{6}$ are the minimal generators of $\operatorname{in}(I,g)$. This means that the $4$ first columns of $\Gamma_{1,6}$ are linearly independent, so that $\mathbf{B}_{6}^{S_{1}}=\mathbf{B}_{6}^{[1,4]}$. Thus, the set of generators of $\operatorname{in}(I,g)$ as in Proposition 3.3 is

[TABLE]

Let $F=B(5;2,3,3,4,5)$ be the set of the standard monomials with respect to $(I,g)$. Denote by $f_{i}=|F_{i}|$ and $f^{\prime}_{i}=|\widetilde{F}_{i}^{0}|$. By Corollary 3.8, we have*

---

[TABLE]

In [4, Lemma 3.14], it is conjectured that $\mathbf{B}_{d+i}^{S_{i}}$ are the $a_{i}$ largest monomials of $\mathbf{B}_{d+i}$ for every $i=0,\cdots,i^{*}$. However, in the following example, we show that this conjecture is not true.

Example 3.10**.**

Let $(I,g)=(f_{1},\cdots,f_{5},g)$ be the generic ideal of type $(6;2,3,3,4,5,5)$ in $K[x_{1},\cdots,x_{5},z]$. Let $F=B(5;2,3,3,4,5)$ as in Example 3.9 with the variable $x_{5}$ instead of variable $z$. Then $\delta=12,\sigma=6$, $d=5$ and $i^{*}=\lfloor\frac{\delta-d}{2}\rfloor=3$. Here $F$ plays the same role of $B$ in Proposition 3.3. We write**

[TABLE]

For $i=0$, The largest monomials of $\mathbf{F}_{5}$ will be added, in this case it is $x_{2}^{2}x_{3}x_{4}^{2}$.**

For $i=1,$

[TABLE]

where $\mathbf{F}_{1}=\mathbf{\widetilde{F}}_{1}^{0}\cup\{x_{5}\}$ and $\mathbf{F}_{6}=\mathbf{\widetilde{F}}_{6}^{0}\cup x_{5}\mathbf{\widetilde{F}}_{5}^{0}\cdots\cup x_{5}^{5}\mathbf{\widetilde{F}}_{1}^{0}\cup\{x_{5}^{6}\}$. The monomials in $\mathbf{F}_{6}$ corresponding to the $f_{1}=5$ first linearly independent columns of $\Gamma_{1,6}$ will be added. We have**

[TABLE]

Since $|\mathbf{\widetilde{F}}_{1}^{0}|=f^{\prime}_{1}=4$ and $|\mathbf{\widetilde{F}}_{6}^{0}|=f^{\prime}_{6}=5$, we get $\operatorname{rank}(\Omega_{1,6})\leq 4$. By using Macaulay2 to compute $\operatorname{in}(I,g)$, we see that the $4$ largest monomials of $\mathbf{F}_{6}$ are the minimal generators of $\operatorname{in}(I,g)$. This means that the $4$ first columns of $\Omega_{1,6}$ are linearly independent. Hence the $5$ first linearly independent columns of $\Gamma_{1,6}$ are the $4$ first columns and the $6$-th column. Thus $\mathbf{F}_{6}^{S_{1}}=\mathbf{F}_{6}^{[1,4]}\cup\mathbf{F}_{6}^{\{6\}}$. However the monomial $\mathbf{F}_{6}^{\{6\}}$ is not a minimal generator because $\mathbf{F}_{6}^{\{6\}}=x_{5}\mathbf{F}_{5}^{\{1\}}$ and $\mathbf{F}_{5}^{\{1\}}$ was already added in step $i=0$.**

4 Application for Fröberg’s Conjecture

In [17, Theorem 2], Pardue proved that Fröberg’s Conjecture is equivalent to Conjecture 1.4. In order to prove the equivalence of the conjectures, Pardue used the notion of semi-regular sequences already quoted in the introduction and introduced in [17, Section 3]. Regular sequences and semi-regular sequences can be characterized by Hilbert series.

Proposition 4.1**.**

[17, Proposition 1]** Let $A=K[x_{1},\cdots,x_{n}]/I$, where $I$ is a homogeneous ideal, and $f_{1},\cdots,f_{r}$ are homogeneous polynomials of degree $d_{1},\cdots,d_{r}$. Then, (1) $f_{1},\cdots,f_{r}$ is a semi-regular sequence on $A$ if and only if for all $s=1,\cdots,r$

[TABLE]

(2) $f_{1},\cdots,f_{r}$ is a regular sequence on $A$ if and only if

[TABLE]

In [17, Theorem 2], Pardue proved that Conjecture 1.4 is equivalent to the following conjecture.

Conjecture 4.2**.**

[17, Conjecture C]** Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$. Then $x_{n},x_{n-1},\cdots,x_{1}$ is a semi-regular sequence on $A=K[x_{1},\cdots,x_{n}]/I$.

We apply now Theorem 3.1 to get partial answers to Conjecture 1.4 and Conjecture 4.2. Let $d_{1}\leq\cdots\leq d_{n}$ be $n$ positive integers. For every $1\leq i\leq n$, we set                                   $\delta_{i}=d_{1}+\cdots+d_{i}-i,$                                   $\sigma_{i}=\min\left\{\delta_{i-1},\left\lfloor\dfrac{\delta_{i}}{2}\right\rfloor\right\}$ for all $i\geq 2$.

Theorem 4.3**.**

Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$ with $n\leq 3$ and $d_{1}\leq\cdots\leq d_{n}$. Then, Conjecture 1.4 is true for $I$.

Proof.

It is known that Conjecture 1.4 is true in case $n\leq 2$. For $n=3$, we have $J=\pi(I)$ is a generic ideal of type $(2;d_{1},d_{2})$. Hence, Conjecture 1.4 is true for $J$. Since

[TABLE]

by Theorem 3.1 we have that Conjecture 1.4 is true for $I$. ∎

Theorem 4.4**.**

Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$ with $n\geq 4$ and $d_{1}\leq\cdots\leq d_{n}$. If $d_{i}\geq\sigma_{i-1}$ for all $4\leq i\leq n$, then Conjecture 1.4 is true for $I$.

Proof.

We prove by induction on $n$. For $n=4$, we have $J=\pi(f_{1},f_{2},f_{3})$ is a generic ideal of type $(3;d_{1},d_{2},d_{3})$. By Theorem 4.3, Conjecture 1.4 is true for $J$. Since $d_{4}\geq\sigma_{3}$, by Theorem 3.1 we have that Conjecture 1.4 is true for $I$.

For $n>4$, we have $J=\pi(f_{1},\cdots,f_{n-1})$ is a generic ideal of type $(n-1;d_{1},\cdots,d_{n-1})$ with $d_{i}\geq\sigma_{i-1}$ for all $4\leq i\leq n-1$. Hence, by induction Conjecture 1.4 is true for $J$. Since, $d_{n}\geq\sigma_{n-1}$, by Theorem 3.1 we have that Conjecture 1.4 is true for $I$. ∎

Since Conjecture 1.4 is equivalent to Conjecture 4.2, we also get a partial answer to Conjecture 4.2.

Theorem 4.5**.**

Let $I=(f_{1},\cdots,f_{n})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{n})$ in $K[x_{1},\cdots,x_{n}]$ with $d_{1}\leq\cdots\leq d_{n}$. If $d_{i}\geq\sigma_{i-1}$ for all $4\leq i\leq n$ (in case $n\geq 4$), then $x_{n},x_{n-1},\cdots,x_{1}$ is a semi-regular sequence on $K[x_{1},\cdots,x_{n}]/I$.

We apply now Theorem 4.5 to prove the main result of this section.

Theorem 4.6**.**

Let $I=(f_{1},\cdots,f_{r})$ be a generic homogeneous ideal of type $(n;d_{1},\cdots,d_{r})$ in $R=K[x_{1},\cdots,x_{n}]$ with $d_{1}\leq\cdots\leq d_{r}$. If $d_{i}\geq\sigma_{i-1}$ for all $4\leq i\leq r$ (in case $r\geq 4$), then the Hilbert series of $R/I$ is given by

[TABLE]

Proof.

Since Fröberg’s Conjecture is known to be true if $r\leq n$, we only have to consider the case $r>n$. Let $R^{\prime}=K[x_{1},\cdots,x_{r}]$ be the polynomial ring in $r$ variables and view $R$ as $R=R^{\prime}/(x_{r},\cdots,x_{n+1})$. Then, there exist the generic homogeneous polynomials $f_{1}^{\prime},\cdots,f_{r}^{\prime}$ of type $(r;d_{1},\cdots,d_{r})$ in $R^{\prime}$ such that $f_{i}$ is the image of $f_{i}^{\prime}$ in $R=R^{\prime}/(x_{r},\cdots,x_{n+1})$. Set $A=R^{\prime}/(f_{1}^{\prime},\cdots,f_{r}^{\prime})$. It is known that $A$ is the complete intersection and Hilbert series of $A$ is given by

[TABLE]

Applying Theorem 4.5 for $(f_{1}^{\prime},\cdots,f_{r}^{\prime})$ we have $x_{r},\cdots,x_{n+1},\cdots,x_{1}$ is a semi-regular sequence on $A$. By Proposition 4.1 we get

[TABLE]

and Theorem 4.6 follows from the following isomorphisms.

[TABLE]

∎

Acknowledgements

I thank my advisor Maria Evelina Rossi for suggesting the problem and for providing helpful suggestions throughout the preparation of this manuscript. I am also grateful to the department of Mathematics of Genova University for supporting my PhD program. The main algorithm that I used in this paper is the incremental method from [12] and [4]. So, I also wish to thank Gao, Guan, Volny and Capaverde for providing this effective approach.