# On schemes evinced by generalized additive decompositions and their
regularity

Alessandra Bernardi, Alessandro Oneto, Daniele Taufer  Università di Trento,
Via Sommarive, 14 - 38123 Povo (Trento), Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>KU Leuven<EMAIL_ADDRESS>
###### Abstract.

We define and explicitly construct schemes evinced by generalized additive
decompositions (GADs) of a given $d$-homogeneous polynomial $F$. We employ
GADs to investigate the regularity of $0$-dimensional schemes apolar to $F$,
focusing on those satisfying some minimality conditions. We show that
irredundant schemes to $F$ need not be $d$-regular, unless they are evinced by
special GADs of $F$. Instead, we prove that tangential decompositions of
minimal length are always $d$-regular, as well as irredundant apolar schemes
of length at most $2d+1$.

###### Key words and phrases:

Generalized additive decompositions, $0$-dimensional schemes, Hilbert
function, regularity, cactus rank

###### 2020 Mathematics Subject Classification:

14N07, 13D40

##  1\. Introduction

Algebraic and geometric properties of $0$-dimensional schemes have been
largely studied from several perspectives in algebraic geometry, commutative
algebra, and computational algebra. Through _apolarity theory_ , these studies
find applications in the study of _additive decompositions_ of homogeneous
polynomials and, more in general, _tensor decompositions_ [Lan12, BCC+18,
BC19].

In this paper, we are interested in $0$-dimensional schemes that are _apolar_
to a given $d$-homogeneous polynomial $F$, namely the $0$-dimensional schemes
defined by ideals annihilating $F$ by derivation. Understanding the possible
Hilbert functions of minimal apolar schemes is a deep and largely open
question, which could give useful information on the nature of additive
decompositions of polynomials and secant varieties, and whose grasp is
challenging even in moderately small cases [RS000, IR01, BB12, ER12, CCG12,
LO13, BBT13, BB14b, CJN15, Jel18, BJMR18, Chi19, MO20, BB21, ACO23].

Our work aims to study when these Hilbert functions stabilize, and more
specifically at discerning essential conditions for a given $d$-homogeneous
polynomial to have _minimal_ $0$-dimensional apolar schemes that are regular
in degree $d$. This subtle problem carries far-reaching implications spanning
the domains of classical algebraic geometry and complexity theory. In the
context of algebraic geometry, these concepts are part of a longstanding
tradition of exploring secant varieties and Waring problems, see [BCC+18] for
a general overview. From a complexity theory perspective, the knowledge of the
regularity of minimal apolar schemes to a given polynomial might improve the
efficiency of symbolic algorithms for computing ranks and minimal
decomposition of polynomials [BCMT10, BT20, BDHM17].

###  1.1. Additive decompositions

As already recalled, the study of apolar schemes is related to notions of
_rank_ and _additive decompositions_ associated with homogeneous polynomials.
The minimal length of a $0$-dimensional scheme apolar to $F$ is the _cactus
rank_ of $F$ [RS11, BB14a]. If we restrict to schemes that are locally
contained in $(d+1)$-fat points, then they correspond to _generalized additive
decompositions_ (GADs) of $F$, namely expressions as

$F=\sum_{i=1}^{r}L_{i}^{d-k_{i}}G_{i},$

where the $L_{i}$’s are pairwise non-proportional linear forms not dividing
the corresponding $G_{i}$’s [IK99, BBM14, BT20]. Special cases of such
decompositions include _tangential decompositions_ , when $k_{i}=1$ [BT20,
CGG, Bal22], and _Waring decompositions_ , when $k_{i}=0$ [Ger96, BCC+18].

This algebraic description of ranks and additive decompositions has a
geometric interpretation in terms of _Veronese varieties_ and their _secant
varieties_ [Zak93, Ådl87, BCC+18]. A Waring decomposition corresponds to a set
of points on the Veronese variety whose linear span contains the projective
point corresponding to the polynomial $F$. Analogously, tangential
decompositions (generalized additive decompositions, respectively) correspond
to a set of points on the tangential variety (osculating varieties,
respectively) of the Veronese variety whose linear span contains the
projective point of $F$ [BCC+18, CGG, BCGI07, BCGI09, BF03]. In this view,
GADs parameterize generic points of a _joint variety_ of osculating varieties
to certain Veronese variety.

###  1.2. Content of the paper and main results

After recalling the standard definition and results in Section 2, we define
and provide an explicit construction of schemes evinced by GADs in Section 3.
This construction locally agrees with the natural apolar schemes defined in
[BJMR18], but is made effective by delving into the computational details. An
implementation of this construction routine in Macaulay2 [GS] and Magma
[BCP97] can be found in [BOT].

In Section 4 we investigate the weaker and more geometric irredundancy
condition, i.e. we look at schemes that are minimal by inclusion among the
apolar schemes to a given form $F$ of degree $d$. With Example 4.4 we observe
that schemes evinced by GADs might well be redundant, whereas we prove in
Proposition 4.3 that irredundant schemes are evinced by a GAD of $F$ precisely
when their connected components are contained in $(d+1)$-fat points.
Therefore, all schemes apolar to $F$ with _short_ components are evinced by
certain families of GADs of $F$. However, Example 4.6 shows that schemes with
_long_ components may only arise from GADs of higher degree polynomials.

In Section 5 we tackle the regularity of minimal apolar schemes. We show that
non-redundancy to a degree-$d$ form is not enough to ensure $d$-regularity.
Indeed, in Examples 5.8 and 5.10 we present degree-$d$ homogenous polynomials
admitting an apolar scheme that is irredundant but not $d$-regular. However,
we notice that in both cases such schemes are not minimal by length.

In Proposition 5.2 we show that the addenda constituting a GAD evincing an
irredundant scheme $Z$ may never appear in its inverse systems. We use this
result in Proposition 5.3 to guarantee $d$-regularity for schemes evinced by
GADs such that the $L_{i}$’s are linearly independent and the $k_{i}$’s are
small enough, regardless of the scheme being minimal. However, we point out in
Remark 5.7 that all the assumptions of Proposition 5.3 are sharp.

Drawing from the intuition that schemes with components of low multiplicity
usually exhibit low regularity, in Proposition 5.9 we prove that minimal
tangential decompositions of degree-$d$ forms always evince $d$-regular
schemes. Example 5.10 shows the condition of having minimal length is
essential, while irredundancy is not enough.

Finally, we show in Proposition 5.11 that if the cactus rank of a degree-$d$
form is not greater than $2d+1$, then non-redundancy is actually enough to
guarantee $d$-regularity. In particular, all the schemes of minimal length
apolar to degree-$d$ forms with length smaller or equal to $2d+1$ are
$d$-regular.

### Acknowledgements

We sincerely thank E. Ballico, W. Buczyńska, J. Buczyński, M.V. Catalisano, C.
Ciliberto and B. Mourrain for fruitful conversations. DT acknowledges the
hospitality of the TensorDec Laboratory during a research stay at the
Department of Mathematics at the University of Trento, where part of the
present work has been conducted.

### Funding

AB has been partially supported by GNSAGA of INDAM. DT has been supported by
the European Union’s H2020 Programme ERC-669891, and by the Research
Foundation - Flanders via the FWO postdoctoral fellowship 12ZZC23N and the
travel grant V425623N. All the authors have been partially supported by the
Thematic Research Programme “Tensors: geometry, complexity and quantum
entanglement”, University of Warsaw, Excellence Initiative – Research
University and the Simons Foundation Award No. 663281.

##  2\. Preliminaries

In this paper, $\Bbbk$ will always be an algebraically closed field of
characteristic $0$. Given $\alpha=(\alpha_{0},\ldots,\alpha_{n})$ and
$\beta=(\beta_{0},\ldots,\beta_{n})$ in $\mathbb{N}^{n+1}$, let
$|\alpha|=\sum_{i=0}^{n}\alpha_{i}$ and $\alpha!=\prod_{i=0}^{n}\alpha_{i}!$.
We write $\alpha\succeq\beta$ if $\alpha_{i}\geq\beta_{i}$ for every $0\leq
i\leq n$. We use the standard short notation
$X^{\alpha}=X_{0}^{\alpha_{0}}\cdots X_{n}^{\alpha_{n}}$.

###  2.1. Apolarity

Let
$\mathcal{S}=\Bbbk[X_{0},\ldots,X_{n}]=\bigoplus_{d\in\mathbb{N}}\mathcal{S}_{d}$
and
$\mathcal{R}=\Bbbk[Y_{0},\ldots,Y_{n}]=\bigoplus_{d\in\mathbb{N}}\mathcal{R}_{d}$
be standard graded polynomial rings, where $\mathcal{S}_{d}$ and
$\mathcal{R}_{d}$ denote the $\Bbbk$-vector spaces of degree-$d$ homogeneous
polynomials. We also write $\mathcal{S}_{\leq d}=\bigoplus_{e\leq
d}\mathcal{S}_{e}$ and $\mathcal{R}_{\leq d}=\bigoplus_{e\leq
d}\mathcal{R}_{e}$.

We consider the apolarity action of $\mathcal{R}$ on $\mathcal{S}$ given by
differentiation, i.e.,

$Y^{\beta}\circ
X^{\alpha}=\begin{cases}\partial_{\beta}(X^{\alpha})=\frac{\alpha!}{(\alpha-\beta)!}X^{\alpha-\beta}&\text{
if }\alpha\succeq\beta,\\\ 0&\text{ otherwise},\end{cases}$

extended by $\Bbbk$-linearity. Given $F\in\mathcal{S}$, we consider its
annihilator

$\textnormal{Ann}(F)=\\{G\in\mathcal{R}~{}:~{}G\circ F=0\\},$

which is an ideal of $\mathcal{R}$.

This action defines a non-degenerate perfect pairing
$\mathcal{R}_{d}\times\mathcal{S}_{d}\to\Bbbk$ for every $d\in\mathbb{N}$.
Given a subspace $V\subseteq\mathcal{S}_{d}$, we denote by
$V^{\perp}\subseteq\mathcal{R}_{d}$ its orthogonal space with respect to such
pairing. If $V=\langle F\rangle$, we simply denote its orthogonal space by
$F^{\perp}$.

######  Remark 2.1.

A classical result by Macaulay [Mac16] shows that graded Artinian Gorenstein
algebras are all, and only, quotient rings of polynomial rings by annihilator
ideals of homogeneous polynomials, see [Ger96, Theorem 8.7], [IK99, Lemma
2.12] or [Eis13, Theorem 21.6].

In the following, we always identify $\mathcal{R}$ with the coordinate ring of
$\mathbb{P}^{n}=\mathbb{P}(\mathcal{S}_{1})$.

######  Definition 2.2.

Let $F\in\mathcal{S}_{d}$. A $0$-dimensional scheme $Z\subset\mathbb{P}^{n}$
is apolar to $F$ if $I(Z)\subseteq\textnormal{Ann}(F)$.

A famous characterization of schemes apolar to a given form is provided by the
well-known Apolarity Lemma, see e.g. [IK99, Lemma 1.15] in the classical case
of reduced schemes, [BJMR18, Lemma 1] for non-reduced scheme or [RGV18, Lemma
1.3] into a more general framework.

######  Lemma 2.3 (Apolarity Lemma).

Let $F\in\mathcal{S}_{d}$. The following are equivalent:

  * •

$F\in I(Z)^{\perp}_{d}$;

  * •

$I(Z)\subset\textnormal{Ann}(F)$.

Let $\mathcal{S}_{\rm dp}$ be the polynomial ring $\mathcal{S}$ equipped with
a divided power structure, i.e. endowed with the divided powers monomial basis
$X^{[\alpha]}=\frac{1}{\alpha!}X^{\alpha}$. We denote by $F_{\rm
dp}\in\mathcal{S}_{\rm dp}$ the polynomial $F\in\mathcal{S}$ expressed in
divided powers.

For convenience in our computation throughout the paper, we also consider the
action of $\mathcal{R}$ on $\mathcal{S}_{\rm dp}$ by contraction, namely,

$Y^{\beta}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}X^{\alpha}=\begin{cases}X^{\alpha-\beta}&\text{
if }\alpha\succeq\beta,\\\ 0&\text{ otherwise}.\end{cases}$

For a given $F\in\mathcal{S}_{\rm dp}$, its annihilator with respect to this
action will be denoted by

$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(F)=\left\\{G\in\mathcal{R}~{}:~{}G\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}F=0\right\\}.$

One can directly verify that
$G\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}F_{\rm
dp}=(G\circ F)_{\rm dp}$.

###  2.2. Minimality

In this paper, we consider the $0$-dimensional schemes apolar to a given
$F\in\mathcal{S}_{d}$. Among them, we are particularly interested in those
that are minimal by inclusion or length.

######  Definition 2.4.

Let $Z\subset\mathbb{P}^{n}$ be a $0$-dimensional scheme apolar to
$F\in\mathcal{S}_{d}$. We say that $Z$ is irredundant to $F$ if there is no
strict subscheme $Z^{\prime}\subsetneq Z$ among the schemes apolar to $F$.

The minimal length of a $0$-dimensional scheme apolar to $F$ is called both
scheme length of $F$ [IK99] or cactus rank of $F$ [BR13, BB14a].

######  Definition 2.5.

Let $Z\subset\mathbb{P}^{n}$ be a $0$-dimensional scheme apolar to
$F\in\mathcal{S}_{d}$. We say that $Z$ evinces the cactus rank, or evinces the
scheme length of $F$, or simply is minimal apolar to $F$, if $Z$ is of minimal
length among the $0$-dimensional schemes in $\mathbb{P}^{n}$ and apolar to
$F$.

###  2.3. Regularity

We study when the Hilbert function of minimal apolar schemes stabilizes.

######  Definition 2.6.

Given a homogeneous ideal $I\subset\mathcal{R}$, the Hilbert function of the
quotient $\mathcal{R}/I$ is the function
$\mathrm{HF}_{\mathcal{R}/I}:\mathbb{N}\to\mathbb{N}$ such that
$\mathrm{HF}_{\mathcal{R}/I}(i)=\dim\mathcal{R}_{i}/I_{i}$, where
$I_{i}=I\cap\mathcal{R}_{i}$. For a scheme $Z\subset\mathbb{P}^{n}$ we denote
the Hilbert function of $Z$ as
$\mathrm{HF}_{Z}=\mathrm{HF}_{\mathcal{R}/I(Z)}$.

We simply write $\mathrm{HF}_{Z}=(a_{0},a_{1},a_{2},\dots)$ to denote
$\mathrm{HF}_{Z}(i)=a_{i}$.

The Hilbert function of a $0$-dimensional scheme $Z$ is always strictly
increasing until it reaches its length ${\rm len}(Z)$, and then it remains
constant.

######  Definition 2.7.

Given a $0$-dimensional scheme $Z\subset\mathbb{P}^{n}$, the regularity of $Z$
is

${\rm
reg}(Z)=\min_{i\in\mathbb{N}}\\{\mathrm{HF}_{Z}(i)=\mathrm{HF}_{Z}(i+1)={\rm
len}(Z)\\}.$

We say that $Z$ is regular in degree $d$, or $d$-regular, if ${\rm reg}(Z)\leq
d$.

##  3\. Schemes evinced by GADs

We devote the present section to connecting two well-known concepts such as
natural apolar schemes [BJMR18] and generalized additive decomposition [IK99].
Their link serves as the cornerstone of our paper, while their explicit
construction may be beneficial even for expert readers. A complete
implementation in Macaulay2 [GS] and Magma [BCP97] of these procedures may be
found in [BOT].

###  3.1. Natural apolar scheme to $F$ supported at $L$

There is a natural way to associate a local scheme apolar to a given
$F\in\mathcal{S}_{d}$ supported at a prescribed point $[L]\in\mathbb{P}^{n}$
[BJMR18, Section 4]. Let $f_{L}\in\mathcal{S}_{\rm
dp}/(L-1)=\underline{\mathcal{S}}_{\rm dp}$ be the dehomogenization of $F_{\rm
dp}$ by $L$. We consider the projection $\mathcal{S}_{\rm
dp}\to\underline{\mathcal{S}}_{\rm dp}$ and its dual projection
$\mathcal{R}\to\underline{\mathcal{R}}$. We denote the latter projection of an
ideal $J\subset\mathcal{R}$ by $\underline{J}\subset\underline{\mathcal{R}}$.
We will always use lowercase letters for the elements and the variables after
these projections, e.g., we identify $\underline{\mathcal{S}}_{\rm
dp}\simeq\Bbbk[x_{1},\ldots,x_{n}]_{\rm dp}$ and
$\underline{\mathcal{R}}\simeq\Bbbk[y_{1},\ldots,y_{n}]$.

######  Definition 3.1.

Let $F\in\mathcal{S}_{d}$ and $L\in\mathcal{S}_{1}$. We define the natural
apolar scheme to $F$ supported at $L$ the scheme
$Z_{F,L}\subset\mathbb{P}^{n}$ supported at $[L]\in\mathbb{P}^{n}$ and locally
defined by
$\underline{I}(Z_{F,L})=\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})\subset\underline{\mathcal{R}}$.

Note that $\underline{\mathcal{R}}$ can be regarded as the coordinate ring of
the affine chart $U_{0}=\\{[L]~{}:~{}Y_{0}\circ L\neq
0\\}\subset\mathbb{P}^{n}$ and $Z_{F,L}$ is a local $0$-dimensional scheme
supported at the origin of $U_{0}$.

Contraction behaves well with dehomogenization with respect to dual variables.
In particular, if $g\in\underline{\mathcal{R}}$ is the dehomogenization of
$G\in\mathcal{R}$ with respect to $Y_{0}$, and
$g\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}f_{X_{0}}=0$,
then
$G\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}F_{\rm
dp}=0$ [BJMR18, Corollary 3], and the last equality implies that $G\circ F=0$
as observed in Section 2.1. Hence, the scheme $Z_{F,L}$ is apolar to $F$
according to Definition 2.2.

######  Lemma 3.2 ([BJMR18, Corollary 4]).

The scheme $Z_{F,L}$ is apolar to $F$.

Here we detail how to concretely construct the ideal defining such a scheme.

Fix $F\in\mathcal{S}_{d}$ and
$L=\ell_{0}X_{0}+\ldots+\ell_{n}X_{n}\in\mathcal{S}_{1}$. Without loss of
generalities we may assume $\ell_{0}=1$. Over $\mathcal{S}$, we consider the
change of variables given by

(1) $\phi:\mathcal{S}\rightarrow\mathcal{S},\qquad\begin{cases}X_{0}\mapsto
X_{0}-\sum_{i=1}^{n}\ell_{i}X_{i},\\\ X_{i}\mapsto X_{i},&\text{ for
}i\in\\{1,\ldots,n\\}.\end{cases}$

We have $\phi(L)=X_{0}$ and $\tilde{F}=\phi(F)$, therefore we can represent
$f_{L}$ as $\tilde{f}_{X_{0}}=\tilde{F}_{\rm
dp}(1,x_{1},\ldots,x_{n})\in\underline{\mathcal{S}}_{\rm dp}$. Then
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})$
is the kernel of the infinite-dimensional _Hankel operator_ [BT20, BCMT10]:

$H(f_{L}):\underline{\mathcal{R}}\to\underline{\mathcal{S}}_{\rm dp},\quad
g\mapsto
g\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}f_{L}.$

However, since
$y^{\beta}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}f_{L}=0$
for every $|\beta|>\deg(f_{L})$, the annihilator of $f_{L}$ is generated by
the kernel of a truncated Hankel operator. Let $e=\deg(f_{L})$ and consider
the restriction

$H^{e+1}(f_{L}):\underline{\mathcal{R}}_{\leq
e+1}\to\left(\underline{\mathcal{S}}_{\rm dp}\right)_{\leq e}.$

Then,
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})=\ker
H^{e+1}(f_{L}).$

Note that the coefficients of the Hankel matrix can be computed directly from
$\tilde{F}$. Indeed, if we label rows and columns of $H^{e+1}(f_{L})$
accordingly to the divided powers monomial basis of
$\left(\underline{\mathcal{S}}_{\rm dp}\right)_{\leq e}$ and the standard
monomial basis of $\underline{\mathcal{R}}_{\leq e+1}$, respectively, we have

(2) $[H^{e+1}(f_{L})]_{\alpha,\beta}={\rm
eval}_{(0,\ldots,0)}\left(y^{\alpha+\beta}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}f_{L}\right)=\begin{cases}Y^{(d-(|\alpha|+|\beta|),\alpha_{1}+\beta_{1},\cdots,\alpha_{n}+\beta_{n})}\circ\tilde{F}&\text{
if }|\alpha|+|\beta|\leq d,\\\ 0&\text{ otherwise.}\end{cases}$

######  Remark 3.3.

Let $g_{\rm dp}\in\underline{\mathcal{S}}_{\rm dp}$ be a degree-$d$ polynomial
obtained from $g\in\underline{\mathcal{S}}$ by passing to divided powers. The
ideal
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g_{\rm
dp})=\textnormal{Ann}(g)$ has minimal generators in degree $d+1$ if and only
if $g$ is a pure $d$-th power. When it is the case, we actually need to
consider the kernel of $H^{e+1}(g_{\rm dp})$ to compute
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g_{\rm
dp})$, see e.g. Example 3.10. However, whenever $g$ is not a pure power, we
may compute its annihilator by restricting its Hankel matrix to $H^{e}(g_{\rm
dp}):\underline{\mathcal{R}}_{\leq e}\to\left(\underline{\mathcal{S}}_{\rm
dp}\right)_{\leq e}$, which makes the kernel computation more efficient, see
e.g. Examples 3.8 and 3.9.

The homogenization
$\tilde{I}=I(Z_{\tilde{F},X_{0}})=[\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})]^{\rm
hom}\subset\mathcal{R}$ with respect to $Y_{0}$ defines a $0$-dimensional
scheme apolar to $\tilde{F}$ and supported at $[X_{0}]\in\mathbb{P}^{n}$ as in
Definition 3.1. Note that the ideal homogenization is the only step in which
non-linear algebra (e.g. Gröbner bases) may be required.

Finally, to obtain the ideal defining $Z_{F,L}$ as in Definition 3.1, we need
to support $\tilde{I}$ on $[L]\in\mathbb{P}^{n}$, hence we perform the change
of coordinate in $\mathcal{R}$ given by the dualization of the inverse of eq.
1:

(3)
$\psi=(\phi^{-1})^{T}:\mathcal{R}\rightarrow\mathcal{R},\qquad\begin{cases}Y_{0}\mapsto
Y_{0},\\\ Y_{i}\mapsto\ell_{i}Y_{0}+Y_{i},&\text{ for
}i\in\\{1,\ldots,n\\}.\end{cases}$

The ideal $I=\psi(\tilde{I})\subset\mathcal{R}$ defines a $0$-dimensional
scheme which is supported at $[L]$ and apolar to $F$. Indeed, the following
lemma shows that the action by derivation is preserved under the changes of
coordinates given by eqs. 1 and 3.

######  Lemma 3.4.

Let $\phi$ and $\psi$ be changes of coordinates of eqs. 1 and 3. Then we have

$\psi(Y^{\beta})\circ\phi^{-1}(X^{\alpha})=Y^{\beta}\circ X^{\alpha}.$

###### Proof.

We write

$\psi(Y^{\beta})\circ\phi^{-1}(X^{\alpha})=\psi(Y_{0}^{\beta_{0}})\circ\left(\psi(Y_{1}^{\beta_{1}})\circ\left(\dots\psi(Y_{n}^{\beta_{n}})\circ\phi^{-1}(X^{\alpha})\right)\right).$

By the chain rule of derivation, if $L\circ M=0$ then $L^{b}\circ M^{a}=0$ for
any $a,b\in\mathbb{N}$. In particular, for every $j\in\\{1,\dots,n\\}$ we have

$\displaystyle\psi(Y_{j}^{\beta_{j}})\circ\phi^{-1}(X^{\alpha})$
$\displaystyle=(-\ell_{j}Y_{0}+Y_{j})^{\beta_{j}}\circ\left[\left(X_{0}+\ell_{1}X_{1}+\ldots+\ell_{n}X_{n}\right)^{\alpha_{0}}\prod_{i=1}^{n}X_{i}^{\alpha_{i}}\right]$
$\displaystyle=\left[(X_{0}+\ell_{1}X_{1}+\ldots+\ell_{n}X_{n})^{\alpha_{0}}\prod_{\begin{subarray}{c}1\leq
i\leq n\\\ i\neq
j\end{subarray}}X_{i}^{\alpha_{i}}\right]\cdot(Y_{j}^{\beta_{j}}\circ
X_{j}^{\alpha_{j}}).$

Therefore, by repeatedly applying the above equation for every $j$ we obtain

$\psi(Y_{1}^{\beta_{1}})\circ\left(\dots\psi(Y_{n}^{\beta_{n}})\circ\phi^{-1}(X^{\alpha})\right)=(X_{0}+\ell_{1}X_{1}+\ldots+\ell_{n}X_{n})^{\alpha_{0}}\cdot(Y_{1}^{\beta_{1}}\circ
X_{1}^{\alpha_{1}})\cdots(Y_{n}^{\beta_{n}}\circ X_{n}^{\alpha_{n}}).$

The result follows by acting with $\psi(Y_{0}^{\beta_{0}})=Y_{0}^{\beta_{0}}$
on the above quantity. ∎

Note that our choice of the change of coordinates in eq. 1 was arbitrary. It
would have been enough to consider any set of linear forms
$\\{L_{1},\ldots,L_{n}\\}$ completing a basis of $\mathcal{S}_{1}$ together
with $L$. Then, $\phi$ is the change of coordinates inverse to the one sending
$X_{0}\mapsto L_{0}$ and $X_{i}\mapsto L_{i}$, for any $i\in\\{1,\ldots,n\\}.$

######  Algorithm 1 (Natural Apolar Scheme).

Summary of construction of natural apolar schemes.

Input: A homogeneous polynomial $F\in\mathcal{S}_{d}$ and a linear form
$L\in\mathcal{S}_{1}$.
Output: The ideal $I(Z_{F,L})\subseteq\mathcal{R}$.

  1. (1)

Define $\tilde{F}$ as the base-change of $F$ as in eq. 1.

  2. (2)

Compute $f_{L}$ as $\tilde{F}_{\rm dp}(1,x_{1},\dots,x_{n})$ and set
$e=\deg(f_{L})$.

  3. (3)

Compute the ideal $\underline{I}=\ker H^{e+1}(f_{L})$.

  4. (4)

Compute the homogenization $I\subset\mathcal{R}$ of
$\underline{I}\subset\underline{\mathcal{R}}$.

  5. (5)

Return the base-change of the ideal $I$ as in eq. 3.

###  3.2. Generalized Additive Decompositions (GADs)

We recall the definition of the so-called generalized additive decompositions
as introduced in [IK99], and we associate to them $0$-dimensional schemes by
employing the notion of natural apolar scheme introduced in Section 3.1.

######  Definition 3.5.

Let $F\in\mathcal{S}_{d}$ and let $L_{1},\ldots,L_{s}\in\mathcal{S}_{1}$ be
pairwise non-proportional linear forms. A generalized additive decomposition
(GAD) of $F$ supported at $\\{L_{1},\ldots,L_{s}\\}$ is an expression

(4) $F=\sum_{i=1}^{s}L_{i}^{d-k_{i}}G_{i},$

where for every $i\in\\{1,\ldots,s\\}$ we have $0\leq k_{i}\leq d$ and
$G_{i}\in\mathcal{S}_{k_{i}}$ is not divisible by $L_{i}$. If $s=1$, we call
this GAD _local_.

Following [BBM14, BJMR18], we associate a $0$-dimensional scheme to any GAD as
eq. 4.

######  Definition 3.6.

The scheme evinced by a GAD as in eq. 4 is the union of the natural apolar
schemes to each summand with respect to the corresponding $L_{i}$, i.e.,

$Z=\bigcup_{i=1}^{s}Z_{L_{i}^{d-k_{i}}G_{i},L_{i}}.$

The size of a GAD as in eq. 4 is the length of the evinced scheme $Z$.

Note that the same scheme may be evinced by different GADs. Indeed, $L^{d-k}G$
and $L^{d-k}G^{\prime}$ evince the same scheme whenever
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g_{L})=\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g^{\prime}_{L})$.
However, schemes evinced by GADs of a given $F$ are always apolar to it.

######  Lemma 3.7.

Let $Z$ be the scheme evinced by a GAD of $F$. Then $Z$ is apolar to $F$.

###### Proof.

To ease notation, denote $F_{i}=L_{i}^{d-k_{i}}G_{i}$ in eq. 4. Let
$I(Z)_{d}=I(Z)\cap\mathcal{R}_{d}$ and let $I(Z)_{d}^{\perp}$ be the
orthogonal space via the non-degenerate pairing
$\mathcal{R}_{d}\times\mathcal{S}_{d}\to\Bbbk$ induced by derivation. Then,

$I(Z)^{\perp}_{d}=\left(I\left(Z_{F_{1},L_{1}}\right)_{d}\cap\ldots\cap
I\left(Z_{F_{s},L_{s}}\right)_{d}\right)^{\perp}=I\left(Z_{F_{1},L_{1}}\right)^{\perp}_{d}+\ldots+I\left(Z_{F_{s},L_{s}}\right)_{d}^{\perp},$

see e.g. [Ger96, Proposition 2.6]. For every $i\in\\{1,\ldots,s\\}$ we have
$F_{i}\in I\left(Z_{F_{i},L_{i}}\right)_{d}^{\perp}$ by Lemma 3.2. Hence,
$F\in I(Z)_{d}^{\perp}$ and, by the Apolarity Lemma (Lemma 2.3), this implies
that $I(Z)\subseteq\textnormal{Ann}(F)$. ∎

The ideal defining schemes evinced by GADs can be easily computed by
intersecting the ideals defining natural apolar schemes to local pieces of the
additive decomposition, computed as in 1.

###  3.3. Examples

Here we illustrate the above construction with some examples.

######  Example 3.8.

Let $F=(X_{0}+3X_{1}-2X_{2})(X_{1}+X_{2})X_{2}\in\mathcal{S}_{3}$ and
$L=X_{0}+3X_{1}-2X_{2}\in\mathcal{S}_{1}$. Following 1 we obtain
$\tilde{F}=X_{0}X_{1}X_{2}+X_{0}X_{2}^{2}\in\mathcal{S}$ by

$X_{0}\leftarrow X_{0}-3X_{1}+2X_{2}.$

In divided powers it becomes $\tilde{F}_{\rm
dp}=X_{0}X_{1}X_{2}+2X_{0}X_{2}^{[2]}$, whose de-homogenization by $X_{0}=1$
is equal to $f_{L}=X_{1}X_{2}+2X_{2}^{[2]}\in\underline{\mathcal{S}}_{\rm
dp}$. Since $X_{1}(X_{2}+X_{2})$ is not a pure power, by Remark 3.3 we
consider the truncation of the Hankel matrix in degree $2=\deg(f_{L})$, i.e.,

$H^{2}(f_{L})=\blockarray{ccccccccccc}&1y_{1}y_{2}y_{1}^{2}y_{1}y_{2}y_{2}^{2}\\\
\block{c(cccccccccc)}1000012\\\ y_{1}001000\\\ y_{2}012000\\\
y_{1}^{2}000000\\\ y_{1}y_{2}100000\\\ y_{2}^{2}200000\\\ \ ,$

whose kernel defines the ideal
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})=\big{(}y_{2}(2y_{1}-y_{2}),y_{1}^{2}\big{)}\subset\underline{\mathcal{R}}$.
Its homogenization in $\mathcal{R}$ is the ideal
$\big{(}Y_{2}(2Y_{1}-Y_{2}),Y_{1}^{2}\big{)}$, which we need to base-change as
in eq. 3, i.e.

$Y_{1}\leftarrow-3Y_{0}+Y_{1},\quad Y_{2}\leftarrow 2Y_{0}+Y_{2}.$

This way we obtain the ideal
$I=\left((2Y_{0}+Y_{2})(8Y_{0}-2Y_{1}+Y_{2}),(3Y_{0}-Y_{1})^{2}\right)\subset\mathcal{R}$.
Its radical ideal is $(2Y_{1}+3Y_{2},2Y_{0}+Y_{2})$, i.e. it defines a
$0$-dimensional scheme supported at
$[L]=[X_{1}+3X_{2}-2X_{3}]\in\mathbb{P}^{n}$. One can directly verify that
this scheme has length $4$ and it is apolar to $F$. Indeed, it is immediate to
check that

$\displaystyle(2Y_{0}+Y_{2})(8Y_{0}-2Y_{1}+Y_{2})\circ F$
$\displaystyle=(-16Y_{0}^{2}+4Y_{0}Y_{1}-10Y_{0}Y_{2}+2Y_{1}Y_{2}-Y_{2}^{2})\circ
F=0,$ $\displaystyle(3Y_{0}-Y_{1})^{2}\circ F$
$\displaystyle=(9Y_{0}^{2}-6Y_{0}Y_{1}+Y_{1}^{2})\circ F=0.$

Hence, $I$ is the ideal defining $Z_{F,L}$. $\spadesuit$

######  Example 3.9.

Let $F=(X_{0}+3X_{1}-2X_{2})(X_{1}+X_{2})X_{2}\in\mathcal{S}_{3}$ the same
polynomial of Example 3.8 and consider $L=X_{0}\in\mathcal{S}_{1}$. As the
support is $X_{0}$, we do not need to change coordinates, so we directly de-
homogenize $F_{\rm dp}$ with respect to $L$, obtaining
$f_{L}=y_{1}y_{2}+2y_{2}^{[2]}+6y_{1}^{[2]}y_{2}+2y_{1}y_{2}^{[2]}-12y_{2}^{[3]}$.
Since $F$ is not a pure cube, we consider the truncation of the Hankel matrix
in degree $3=\deg(f_{L})$, namely

$H^{3}(f_{L})=\blockarray{ccccccccccccccc}&1y_{1}y_{2}y_{1}^{2}y_{1}y_{2}y_{2}^{2}y_{1}^{3}y_{1}^{2}y_{2}y_{1}y_{2}^{2}y_{2}^{3}\\\
\block{c(cccccccccccccc)}1000012062-12\\\ y_{1}0010620000\\\
y_{2}01262-120000\\\ y_{1}^{2}0060000000\\\ y_{1}y_{2}1620000000\\\
y_{2}^{2}22-120000000\\\ y_{1}^{3}0000000000\\\ y_{1}^{2}y_{2}6000000000\\\
y_{1}y_{2}^{2}2000000000\\\ y_{2}^{3}-12000000000\\\ \ .$

Its kernel is given by the ideal

$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})=(5y_{2}^{3}+76y_{1}^{2}-12y_{1}y_{2}+36y_{2}^{2},2y_{1}^{2}y_{2}+y_{2}^{3},y_{1}^{3},6y_{1}y_{2}^{2}+y_{2}^{3})\subset\underline{\mathcal{R}}.$

To homogenize it, we compute a Gröbner basis with respect to the graded
lexicographic ordering:

$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{L})=(y_{1}^{3},5y_{1}^{2}y_{2}-38y_{1}^{2}+6y_{1}y_{2}-18y_{2}^{2},15y_{1}y_{2}^{2}-38y_{1}^{2}+6y_{1}y_{2}-18y_{2}^{2},5y_{2}^{3}+76y_{1}^{2}-12y_{1}y_{2}+36y_{2}^{2}).$

Hence the natural apolar scheme is defined by the ideal

$\left(\begin{multlined}Y_{1}^{3},5Y_{1}^{2}Y_{2}-38Y_{0}Y_{1}^{2}+6Y_{0}Y_{1}Y_{2}-18Y_{0}Y_{2}^{2},\\\
15Y_{1}Y_{2}^{2}-38Y_{0}Y_{1}^{2}+6Y_{0}Y_{1}Y_{2}-18Y_{0}Y_{2}^{2},5Y_{2}^{3}+76Y_{0}Y_{1}^{2}-12Y_{0}Y_{1}Y_{2}+36Y_{0}Y_{2}^{2}\end{multlined}Y_{1}^{3},5Y_{1}^{2}Y_{2}-38Y_{0}Y_{1}^{2}+6Y_{0}Y_{1}Y_{2}-18Y_{0}Y_{2}^{2},\\\
15Y_{1}Y_{2}^{2}-38Y_{0}Y_{1}^{2}+6Y_{0}Y_{1}Y_{2}-18Y_{0}Y_{2}^{2},5Y_{2}^{3}+76Y_{0}Y_{1}^{2}-12Y_{0}Y_{1}Y_{2}+36Y_{0}Y_{2}^{2}\right)\subset\mathcal{R}.$

One can easily verify that this ideal indeed defines a $0$-dimensional scheme
apolar to $F$ and supported at $[X_{0}]$, whose length is $6$. $\spadesuit$

######  Example 3.10.

Let $F=(X_{0}+3X_{1}-2X_{2})(X_{1}+X_{2})X_{2}\in\mathcal{S}_{3}$ be the
polynomial of Example 3.8. From the equality
$(X_{1}+X_{2})X_{2}=(\frac{X_{1}}{2}+X_{2})^{2}-(\frac{X_{1}}{2})^{2}$ we
immediately get another non-local GAD of $F$, namely

(5)
$F=\left(\frac{X_{1}}{2}+X_{2}\right)^{2}(X_{0}+3X_{1}-2X_{2})-\left(\frac{X_{1}}{2}\right)^{2}(X_{0}+3X_{1}-2X_{2}).$

We compute the scheme $Z$ evinced by the above GAD, supported at
$[X_{1}+2X_{2}]$ and $[X_{1}]$.

We begin with the first addendum
$F_{1}=\frac{1}{4}(X_{1}+2X_{2})^{2}(X_{0}+3X_{1}-2X_{2})$ and
$L_{1}=X_{1}+2X_{2}$. We can neglect the constant factor $\frac{1}{4}$, and
since $L_{1}$ has no $X_{0}$ terms, we simply switch the roles of $X_{0}$ and
$X_{1}$. In order to de-homogenize with respect to $L_{1}$, we perform the
substitution

$\quad X_{1}\leftarrow X_{1}-2X_{2},$

and we get $(f_{1})_{L_{1}}=x_{0}+3-8x_{2}$. Since $X_{0}+3X_{1}-8X_{2}$ is a
pure power, we need to consider the truncation of the Hankel matrix in degree
$2=\deg\big{(}(f_{1})_{L_{1}}\big{)}+1$, i.e.

$\mathbb{H}^{2}\big{(}(f_{1})_{L_{1}}\big{)}=\blockarray{ccccccc}&1y_{0}y_{2}y_{0}^{2}y_{0}y_{2}y_{2}^{2}\\\
\block{c(cccccc)}1182-16000\\\ y_{0}200000\\\ y_{2}-1600000\\\ \,,$

whose kernel defines the ideal
$(8y_{0}+y_{2},y_{2}^{2})\subset\underline{\mathcal{R}}$. After the
homogenization and the base-change

$Y_{2}\leftarrow-2Y_{1}+Y_{2},$

we obtain the ideal
$\left(8Y_{0}-2Y_{1}+Y_{2},(2Y_{1}-Y_{2})^{2}\right)\subset\mathcal{R}$
defining the scheme $Z_{1}=Z_{F_{1},X_{1}+2X_{2}}$, which is $0$-dimensional,
of length $2$ and supported at the point $[X_{1}+2X_{2}]\in\mathbb{P}^{n}$.

We proceed with the second addendum
$F_{2}=\frac{1}{4}X_{1}^{2}(X_{0}+3X_{1}-2X_{2})$ and $L_{2}=X_{1}$. As above,
$X_{1}$ plays the role of $X_{0}$. Since $(f_{2})_{L_{2}}=x_{0}+3-2x_{2}$, we
again consider the truncation of the Hankel matrix in degree $2$:

$\mathbb{H}^{2}\big{(}(f_{2})_{L_{2}}\big{)}=\blockarray{ccccccc}&1y_{0}y_{2}y_{0}^{2}y_{0}y_{2}y_{2}^{2}\\\
\block{c(cccccc)}1182-4000\\\ y_{0}200000\\\ y_{2}-400000\\\ \,,$

whose kernel defines the ideal
$(2y_{0}+y_{2},y_{2}^{2})\subset\underline{\mathcal{R}}$. Hence, the scheme
$Z_{2}=Z_{F_{2},X_{1}}$ is defined by the ideal
$(2Y_{0}+Y_{2},Y_{2}^{2})\subset\mathcal{R}$, it is $0$-dimensional of length
$2$ and is supported at the point $[X_{1}]\in\mathbb{P}^{n}$. In conclusion,
the GAD of eq. 5 evinces the scheme $Z=Z_{1}\cup Z_{2}$ defined by

$I(Z)=\big{(}8Y_{0}-2Y_{1}+Y_{2},(2Y_{1}-Y_{2})^{2}\big{)}\cap(2Y_{0}+Y_{2},Y_{2}^{2})=(4Y_{0}Y_{1}-10Y_{0}Y_{2}+2Y_{1}Y_{2}-Y_{2}^{2},Y_{0}^{2}).$

One can directly check that $Z$ has length $4$, it is supported at the points
$[X_{1}]$ and $[X_{1}+2X_{2}]$, and it is apolar to $F$. $\spadesuit$

##  4\. GADs and Irredundant schemes

In this section, we investigate irredundant schemes evinced by GADs by
employing ideas on natural apolar schemes from [BJMR18].

######  Remark 4.1.

Let $[L]\in\mathbb{P}^{n}$ be a simple point defined by the ideal
$\wp_{L}\subset\mathcal{R}$. Recall that the $j$-fat point supported at $[L]$
is the $0$-dimensional scheme defined by the ideal $\wp_{L}^{j}$ For any
$k\leq d$, the natural apolar scheme of $F=L^{d-k}G\in\mathcal{S}_{d}$
supported at $[L]$ is contained in the $(k+1)$-fat point supported at $[L]$,
since the localization $f_{L}$ has degree at most $k$. Thus, $Z_{F,L}$ is
$k$-regular, as a $(k+1)$-fat point is always $k$-regular and the containment
preserves the regularity [BCGI07, BCGI09]. Finally, if $F$ is concise in $n+1$
variables, i.e., $\mathrm{HF}_{Z}(1)=n+1$, then $Z_{F,L}$ is regular in degree
$k-n$ since its Hilbert function starts with $\mathrm{HF}_{Z}=(1,n+1,\dots)$
and is strictly increasing until it stabilizes.

######  Remark 4.2.

By [BJMR18, Lemma 3], given a local scheme $Z\subset\mathbb{P}^{n}$ apolar to
$F\in\mathcal{S}_{d}$ and supported at $[L]$, there exists
$G\in\mathcal{S}_{D}$ ($D\geq d$) such that $Z_{G,L}\subseteq Z$ and $F=H\circ
G$ for some $H\in\mathcal{R}_{D-d}$. Furthermore, in [BJMR18, Proposition 1]
it is shown that, under minimality assumption, the localizations of $F_{\rm
dp}$ and $G_{\rm dp}$ with respect to $L$ are equal up to degree $d$. In that
result, the minimality requirement is in terms of minimal length among the
schemes supported at $[L]$ and apolar to $F$. However, we observe that in that
proof irredundancy is actually enough. For the sake of completeness, we report
here the proof of the following statement, which may be seen as a non-local
version of [BJMR18, Proposition 1].

######  Proposition 4.3.

Let $Z$ be a $0$-dimensional scheme apolar and irredundant to
$F\in\mathcal{S}_{d}$. Then $Z$ is evinced by a GAD of $F$ if and only if
there are $L_{1},\dots,L_{s}\in\mathcal{S}_{1}$ such that
$I(Z)\supseteq\bigcap_{i=1}^{s}\wp_{L_{i}}^{d+1}$.

###### Proof.

Let $Z=Z_{1}\cup\dots\cup Z_{s}$ be the irreducible decomposition of $Z$.

If $Z$ is evinced by a GAD as in eq. 4, then each
$Z_{i}=Z_{L_{i}^{d-k_{i}}G_{i}}$ is contained in a $(k_{i}+1)$-fat point by
Remark 4.1, hence
$I(Z_{i})\supseteq\wp_{L_{i}}^{k_{i}+1}\supseteq\wp_{L_{i}}^{d+1}$. Note that
this implication does not need irredundancy.

Conversely, since $I(Z)\subseteq\textnormal{Ann}(F)$, then we have

$F\in I(Z)^{\perp}_{d}=I(Z_{1})_{d}^{\perp}+\dots+I(Z_{s})^{\perp}_{d}.$

Therefore, we have an additive decomposition $F=\sum_{i=1}^{s}F_{i}$ with
$F_{i}\in I(Z_{i})^{\perp}_{d}$. By Remark 4.2 there are
$G_{i}\in\mathcal{S}_{D_{i}}$ and $H_{i}\in\mathcal{R}_{D_{i}-d}$ such that
$Z_{G_{i},L_{i}}\subseteq Z_{i}$ and $H_{i}\circ G_{i}=F_{i}$. By [BJMR18,
Lemma 3] we know that
$h_{i}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}(g_{i})_{L_{i}}$
and $(f_{i})_{L_{i}}$ are equal up to degree $d$, but since
$I(Z_{i})\supseteq\wp_{L_{i}}^{d+1}$, the degree of the local generator
$h_{i}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}(g_{i})_{L_{i}}$
is bounded by $d$, so it equals $(f_{i})_{L_{i}}$. Hence, we have

$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}\big{(}(f_{i})_{L_{i}}\big{)}=\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}\big{(}h_{i}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}(g_{i})_{L_{i}}\big{)}\supseteq\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}\big{(}(g_{i})_{L_{i}}\big{)},$

therefore the natural apolar scheme $Z_{F_{i},L_{i}}$ is contained in
$Z_{G_{i},L_{i}}$ and is apolar to $F_{i}$. However, the scheme $Z_{i}$ needs
to be irredundant to $F_{i}$, hence we conclude that
$Z_{F_{i},L_{i}}=Z_{G_{i},L_{i}}=Z_{i}$. Therefore $Z$ is the scheme evinced
by the additive decomposition $\sum_{i=1}^{s}F_{i}$ supported at
$L_{1},\dots,L_{s}$. ∎

In the following example, we observe that even if $Z$ is evinced by a GAD of
$F\in\mathcal{S}_{d}$ and its components are contained in $(d+1)$-fat points,
$Z$ may still be redundant to $F$.

######  Example 4.4.

Consider the GAD $F=X_{0}G_{1}+X_{1}^{2}G_{2}\in\mathcal{S}_{3}$, where

$G_{1}=4X_{0}^{2}+2X_{0}X_{1}-4X_{1}^{2},\quad G_{2}=-3X_{0}-5X_{1}.$

The scheme $Z$ evinced by such GAD is given by the ideal

$I(Z)=(Y_{0}^{2}Y_{1}^{3})=\wp_{X_{0}}^{3}\cap\wp_{X_{1}}^{2}\subset\mathcal{R}.$

Its Hilbert function is $\mathrm{HF}_{Z}=(1,2,3,4,5,5,\dots)$, hence it is not
$3$-regular. We move the addendum in $G_{1}$ containing $X_{1}^{2}$ to
$G_{2}$, obtaining a different GAD supported at the same points:
$F=X_{0}^{2}\tilde{G}_{1}+X_{1}^{2}\tilde{G}_{2}$, where

$\tilde{G}_{1}=4X_{0}+2X_{1},\quad\tilde{G}_{2}=-7X_{0}-5X_{1}.$

The scheme $\tilde{Z}$ evinced by the last GAD is by construction apolar to
$F$, and it is defined by

$I(\tilde{Z})=(Y_{0}^{2}Y_{1}^{2})=\wp_{X_{0}}^{2}\cap\wp_{X_{1}}^{2}\subset\mathcal{R}.$

The Hilbert function of $\tilde{Z}\neq Z$ is
$\mathrm{HF}_{\tilde{Z}}=(1,2,3,4,4,\dots)$, and clearly $I(Z)\subsetneq
I(\tilde{Z})\subseteq\textnormal{Ann}(F)$. Hence, $Z$ is redundant.

It can be directly verified that $\tilde{Z}$ is irredundant to $F$ (e.g. as in
Example 5.10), but not minimal, since

$I_{W}=(79Y_{0}^{2}-166Y_{0}Y_{1}+88Y_{2}^{2})\subset\mathcal{R}$

is evinced by the (unique) Waring decomposition of $F$, defining a scheme of
length $2$ apolar to $F$. $\spadesuit$

######  Corollary 4.5.

Let $L_{1},\ldots,L_{s}\in\mathcal{S}_{1}$ and $Z=Z_{1}\cup\ldots\cup Z_{s}$
be a $0$-dimensional scheme apolar to $F\in\mathcal{S}_{d}$ such that for
every $i\in\\{1,\dots,s\\}$ we have
$I(Z_{i})\supset\wp_{L_{i}}^{\tilde{k}_{i}+1}$ with $\tilde{k}_{i}\leq d$.
Then $Z$ contains a scheme evinced by a GAD of $F$ as in eq. 4, with
$k_{i}\leq\tilde{k}_{i}$.

###### Proof.

Let $Y=Y_{1}\cup\ldots\cup Y_{s}\subseteq Z$ be non-redundant and apolar to
$F$, with $Y_{i}\subseteq Z_{i}$. Then, it is enough to apply the proof of
Proposition 4.3 to $Y$, since

$I(Y_{i})^{\perp}_{d}\subseteq
I(Z_{i})^{\perp}_{d}\subseteq(\wp_{i}^{\tilde{k}_{i}+1})_{d}^{\perp}=\langle
L_{i}^{d-{\tilde{k}_{i}}}Q~{}:~{}Q\in\mathcal{S}_{\tilde{k}_{i}}\rangle,$

where the last equality is a classical result, see e.g. [Ger96, Theorem 3.2].
We conclude that $I(Y_{i})$ is evinced by
$F_{i}=L_{i}^{d-{\tilde{k}_{i}}}Q_{i}$, which becomes a valid (local) GAD
after collecting all the factors $L_{i}$ in $Q_{i}$. Thus, $Y$ is evinced by
the GAD $F=\sum_{i=1}^{s}F_{i}$ supported at $L_{1},\dots,L_{s}$. ∎

In the following example, we show that the degree $D$ of the polynomial $G$
from Remark 4.2 may well exceed $d=\deg(F)$. We thank J. Buczyński for
pointing it out.

######  Example 4.6.

Consider the following polynomial:

$\displaystyle F=24\,X_{0}^{3}$
$\displaystyle+70\,X_{0}^{2}X_{1}+75\,X_{0}^{2}X_{2}+70\,X_{0}^{2}X_{3}+180\,X_{0}^{2}X_{4}+10\,X_{0}^{2}X_{5}+10\,X_{0}X_{1}^{2}$
$\displaystyle+70\,X_{0}X_{2}^{2}+360\,X_{0}X_{2}X_{3}+120\,X_{0}X_{2}X_{4}+60\,X_{0}X_{3}^{2}+60\,X_{2}^{3}+60\,X_{2}^{2}X_{3}\in\mathcal{S}_{3},$

and let $Z$ be the scheme defined by the ideal

$\displaystyle I(Z)=(-Y_{0}Y_{3}+Y_{2}^{2},\,-Y_{1}Y_{4}+Y_{2}Y_{3},\,$
$\displaystyle-
Y_{1}Y_{5}+Y_{1}^{2},\,-6Y_{1}Y_{5}+Y_{2}Y_{4},\,-6Y_{1}Y_{5}+Y_{3}^{2},$
$\displaystyle
Y_{1}Y_{2},\,Y_{1}Y_{3},\,Y_{1}Y_{4},\,Y_{1}Y_{5},\,Y_{2}Y_{5},\,Y_{3}Y_{4},\,Y_{3}Y_{5},\,Y_{4}^{2},\,Y_{4}Y_{5},\,Y_{5}^{2})\subset\mathcal{R}.$

One can computationally check that $Z$ is a local $0$-dimensional scheme
apolar to $F$, of minimal length $6$ and supported at
$[X_{0}]\in\mathbb{P}^{n}$. One can also verify that it is the unique scheme
of minimal length apolar to such $F$, by explicitly computing minimal apolar
schemes [BT20], or by observing that
$I(Z)=\textnormal{Ann}(F)\cap\mathcal{R}_{\leq 2}$ and the Hilbert function of
$\mathcal{R}/\textnormal{Ann}(F)$ is $(1,6,6,1)$. In particular, $Z$ is non-
redundant. Since $I(Z)\not\supseteq\wp_{X_{0}}^{4}$, by Proposition 4.3 there
is no GAD of $F$ that evinces this apolar scheme. However, as recalled in
Remark 4.2, since $I(Z)\supseteq\wp_{X_{0}}^{5}$ then $Z$ is evinced by a GAD
of a degree-$4$ polynomial $G$ having $F$ among its partials. Indeed, let us
consider the polynomial

$\displaystyle G=\ $ $\displaystyle
6X_{0}^{4}+\frac{70}{3}X_{0}^{3}X_{1}+25X_{0}^{3}X_{2}+\frac{70}{3}X_{0}^{3}X_{3}+60X_{0}^{3}X_{4}+\frac{10}{3}X_{0}^{3}X_{5}+5X_{0}^{2}X_{1}^{2}+35X_{0}^{2}X_{2}^{2}$
$\displaystyle+180X_{0}^{2}X_{2}X_{3}+60X_{0}^{2}X_{2}X_{4}+30X_{0}^{2}X_{3}^{2}+60X_{0}X_{2}^{3}+60X_{0}X_{2}^{2}X_{3}+5X_{2}^{4}\in\mathcal{S}_{4}.$

Note that $Y_{0}\circ G=F$. Moreover, $Z=Z_{G,X_{0}}$, i.e., it is evinced by
the trivial GAD of $G$ given by $G=X_{0}^{0}G$. This example shows why the
containment in $(d+1)$-fat points is crucial for Proposition 4.3 and Corollary
4.5. In particular, we have that

$\displaystyle g_{X_{0}}$ $\displaystyle=120x_{2}^{4}+f_{X_{0}}=$
$\displaystyle=120x_{2}^{4}+360x_{2}^{3}+120x_{2}^{2}x_{3}+20x_{1}^{2}+140x_{2}^{2}+360x_{2}x_{3}+120x_{2}x_{4}+120x_{3}^{2}+140x_{1}+150x_{2}$
$\displaystyle\quad+140x_{3}+360x_{4}+20x_{5}+144.$

We observe that $g_{X_{0}}$ and $f_{X_{0}}$ are equal up to degree $3$, but
since

$(y_{2}^{2}-y_{3})\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}f_{X_{0}}=-120x_{2}^{2}\neq
0,$

then
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g_{X_{0}})\not\subseteq\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(f_{X_{0}})$.
$\spadesuit$

##  5\. Regularity of schemes evicing GADs

###  5.1. Apolar schemes with low multiplicities and independent supports

For a given $L\in\mathcal{S}_{1}$, let $D_{L}=L^{\perp}\cap\mathcal{R}_{1}$
and $D^{e}_{L}\subset\mathrm{Sym}^{e}\mathcal{R}_{1}$ be its $e$-th symmetric
power. We also define the $\Bbbk$-vector spaces

$\mathcal{D}^{e}_{L}(F)=\langle H\circ F~{}:~{}H\in
D^{e}_{L}\rangle\subseteq\mathcal{S}_{d-e},$

and given a vector space $V\subseteq\mathcal{S}_{m}$ and
$H\in\mathcal{S}_{l}$, we write

$H\cdot V=\\{HF~{}:~{}F\in V\\}\subseteq\mathcal{S}_{l+m}.$

With the notation of the previous sections, as in [BJMR18, Remark 3], we have

$I(Z_{F,L})_{d}^{\perp}=\mathbb{P}\left(\sum_{e=0}^{d}L^{e}\cdot\mathcal{D}^{e}_{L}(F)\right)\subset\mathbb{P}(\mathcal{S}_{d}).$

When $F=L^{d-k}G$, from the above equality and the chain rule of derivation we
get

$I(Z_{L^{d-k}G,L})_{d}^{\perp}=\mathbb{P}\left(\sum_{e=0}^{k}L^{d-k+e}\cdot\mathcal{D}^{e}_{L}(G)\right)\subset\mathbb{P}(\mathcal{S}_{d}).$

######  Remark 5.1.

Let $Z=\cup_{i=1}^{s}Z_{i}\subset\mathbb{P}^{n}$ be the irreducible
decomposition of a $0$-dimensional scheme. Then $Z$ is $h$-regular precisely
when $\dim I(Z)_{h}^{\perp}=\deg(Z)=\sum_{i=1}^{s}\deg(Z_{i})$, therefore
there cannot be $\Bbbk$-linear relations involving generators of
$I(Z_{i})_{h}^{\perp}$ for different $i$’s.

If there is a relation between different $I(Z_{i})_{d}^{\perp}$ as in Remark
5.1, the scheme $Z$ is not $d$-regular. However, the following proposition
shows that if such $Z$ is evinced by a GAD of $F\in\mathcal{S}_{d}$ and is
irredundant to it, such a relation cannot involve addenda appearing in that
GAD.

######  Proposition 5.2.

Let $Z$ be the scheme evinced by the GAD
$F=\sum_{i=1}^{s}L_{i}^{d-k_{i}}G_{i}\in\mathcal{S}_{d}$. If, for some
$i\in\\{1,\dots,s\\}$, we have

(6) $L_{i}^{d-k_{i}}G_{i}\in\sum_{1\leq e_{i}\leq
k_{i}}L_{i}^{d-k_{i}+e_{i}}\cdot\mathcal{D}^{e_{i}}_{L_{i}}(G_{i})+\sum_{\begin{subarray}{c}1\leq
j\leq s\\\ j\neq i\end{subarray}}\sum_{0\leq e_{j}\leq
k_{j}}L_{j}^{d-k_{j}+e_{j}}\cdot\mathcal{D}^{e_{j}}_{L_{j}}(G_{j}),$

then $Z$ is redundant to $F$. It is intended that the first sum in eq. 6 is
empty if $k_{i}=0$.

###### Proof.

Without loss of generality, we may assume that in eq. 6 we have $i=1$. We
define a scheme $Z^{\prime}$ apolar to $F$ as follows.

$\bullet$ If $k_{1}=0$, by eq. 6, we simplify the GAD as
$F=\sum_{j=2}^{s}L_{j}^{d-k_{i}}G^{\prime}_{j}$ with
$G^{\prime}_{j}\in\sum_{e=0}^{k_{i}}L_{j}^{e}D^{e}_{L_{j}}(G_{j})$. We call
$Z^{\prime}$ the scheme evinced by this GAD of $F$.

$\bullet$ If $k_{1}>0$, we replace $L_{1}^{d-k_{1}}G_{1}$ in the GAD of $F$
with the linear combination deduced from eq. 6. In particular, there are
elements $H_{j,e_{j}}\in\mathcal{D}_{L_{j}}^{e_{j}}$ and integers
$m_{j}\in\mathbb{N}$ such that we can write

(7) $F=\sum_{j=1}^{s}L_{j}^{d-k_{j}+m_{j}}\left(\sum_{m_{j}\leq e_{j}\leq
k_{j}}L_{j}^{e_{j}-m_{j}}\left(H_{j,e_{j}}\circ G_{j}\right)\right).$

Since $k_{1}>0$, then we have $m_{1}\geq 1$ in eq. 7. The last equation is a
GAD of $F$ up to deleting vanishing addenda and, for all the others, choosing
$m_{j}$ such that $H_{j,m_{j}}\neq 0$. Let $Z^{\prime}$ be the scheme evinced
by the new GAD in eq. 7.

By construction, $Z^{\prime}$ is apolar to $F$, so it is sufficient to show
that $Z^{\prime}\subsetneq Z$.

Following the notation introduced in Section 3.1, let
$g_{j}=(g_{j})_{L_{j}}\in\underline{\mathcal{S}}$ be the de-homogenization of
$(G_{j})_{\rm dp}$ with respect to $L_{j}$, and let
$h_{j,e_{j}}\in\underline{\mathcal{R}}$ be the dehomogenization of
$H_{j,e_{j}}$ with respect to the dual linear form $L_{j}^{*}$ of $L_{j}$.
Since $H_{j,e_{j}}\in D^{e_{j}}_{L_{j}}\subset{\rm Sym}^{e_{j}}L_{j}^{\perp}$,
then $H_{j,e_{j}}$ does not involve $L_{j}^{*}$, so its dehomogenization
$h_{j,e_{j}}$ is equal to $H_{j,e_{j}}$. Thus, the de-homogenization of
$(H_{j,e_{j}}\circ G_{j})_{\rm
dp}=H_{j,e_{j}}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}(G_{j})_{\rm
dp}$ with respect to $L_{j}$ coincides with
$h_{j,e_{j}}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}g_{j}$.
In particular, the $j$-th component of $Z^{\prime}$ is defined by
$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}\big{(}\sum_{m_{j}\leq
e_{j}\leq
k_{j}}h_{j,e_{j}}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}g_{j}\big{)}$.
Since

$\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}\left(\sum_{m_{j}\leq
e_{j}\leq
k_{j}}h_{j,e_{j}}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}g_{j}\right)\supseteq\textnormal{Ann}^{\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}}(g_{j}),$

we deduce that $Z^{\prime}\subseteq Z$. We now show that this containment is
proper.

In the case $k_{1}=0$, then the containment is strict because $Z^{\prime}$ has
no support on $[L_{1}]$. In the case $k_{1}>0$, since $m_{1}\geq 1$,
$\deg(\sum_{m_{i}\leq e_{i}\leq
k_{i}}h_{1,e_{1}}\mathbin{\mathchoice{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\displaystyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\textstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptstyle\lnot$}}}{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\scriptscriptstyle\lnot$}}}}g_{1})<\deg(g_{1})$
so the socle degree of the first component of $I(Z^{\prime})$ and $I(Z)$ are
different, so again they must be different. ∎

######  Proposition 5.3.

Let $s>1$ and $L_{1},\ldots,L_{s}\in\mathcal{S}_{1}$ be $\Bbbk$-linearly
independent forms and $Z$ be the scheme evinced by a GAD of
$F\in\mathcal{S}_{d}$ as in eq. 4. If either

  1. (a)

$d>\max_{i\neq j}\\{k_{i}+k_{j}\\}$, or

  2. (b)

$d>\max_{i\neq j}\\{k_{i}+k_{j}-2\\}$ and $Z$ is irredundant,

then $Z$ is $d$-regular.

###### Proof.

For every $1\leq i\leq s$ we let $Z_{i}$ be the natural apolar scheme to
$F_{i}=L_{i}^{d-k_{i}}G_{i}$ supported at $L_{i}$, so $Z=\cup_{i=1}^{s}Z_{i}$.
By Remark 4.2 each $Z_{i}$ is $d$-regular, therefore $\dim
I(Z_{i})_{d}^{\perp}=\deg(Z_{i})$. By Remark 5.1, we only need to show that
there are no $\Bbbk$-linear relations involving generators of
$I(Z_{i})_{d}^{\perp}$ for different $i$’s. If there was such a relation,
there should exist $Q_{i}\in\sum_{e=0}^{k_{i}}L_{i}^{e}\cdot
D^{e}_{L_{i}}(G_{i})$, for $i=1,\ldots,s$, such that

(8) $L_{i}^{d-k_{i}}Q_{i}=\sum_{i\neq j}L_{j}^{d-k_{j}}Q_{j}.$

Since the $L_{i}$’s are linearly independent, up to a change of coordinates we
can write the above as

$X_{i}^{d-k_{i}}\tilde{Q}_{i}=\sum_{i\neq j}X_{j}^{d-k_{j}}\tilde{Q}_{j}.$

In case (a), the hypothesis $d-k_{i}>k_{j}=\deg(Q_{j})=\deg(\tilde{Q}_{j})$
prevents form factoring $X_{i}^{d-k_{i}}$ out of the right-hand side of the
above equation. Thus, no such relation may hold.

In case (b), since $Z$ is irredundant, by Proposition 5.2 we may assume that
any relation between the $I(Z_{i})_{d}^{\perp}$’s does not involve any of the
terms $L_{i}^{d-k_{i}}G_{i}$. Thus, eq. 8 actually leads to a relation of the
form

$X_{i}^{d-k_{i}+1}\tilde{Q}_{i}=\sum_{i\neq j}X_{j}^{d-k_{j}+1}\tilde{Q}_{j}.$

As in the previous case, the factor $X_{i}^{d-k_{i}+1}$ cannot appear on the
right-hand side of the above sum due to $d>\max_{i\neq j}\\{k_{i}+k_{j}\\}-2$.

In conclusion, in both cases, the $I(Z_{i})_{d}^{\perp}$’s cannot intersect,
so $Z$ is $d$-regular. ∎

######  Remark 5.4.

We note that requiring $s>1$ in Proposition 5.3 is not restrictive, as in the
local case ($s=1$) Remark 4.2 already contains a stronger result.

An immediate corollary of Proposition 5.3 is the following.

######  Corollary 5.5.

Let $Z$ be the scheme evinced by the GAD
$F=\sum_{i=1}^{s}L_{i}^{d-k_{i}}G_{i}\in\mathcal{S}_{d}$, such that
$L_{1},\ldots,L_{s}$ are $\Bbbk$-linearly independent and $k_{i}<\frac{d}{2}$
for every $i\in\\{1,\ldots,s\\}$. Then $Z$ is $d$-regular.

######  Corollary 5.6.

Let $Z=\bigcup_{i=1}^{s}Z_{i}$ be a $0$-dimensional scheme apolar and
irredundant to $F\in\mathcal{S}_{d}$, such that
$I(Z_{i})\supset\wp_{L_{i}}^{\lceil{\frac{d}{2}}\rceil+1}$ and the $L_{i}$ are
$\Bbbk$-linearly independent. Then $Z$ is $d$-regular.

###### Proof.

It follows by Corollary 4.5 together with Corollary 5.5. ∎

######  Remark 5.7.

We notice that every requirement of Proposition 5.3 is sharp. In fact, Example
4.4 shows that the inequality in (a) cannot be improved: if $d=\max_{i\neq
j}\\{k_{i}+k_{j}\\}$ the scheme $Z$ may be not $d$-regular. Similarly, the
following Example 5.8 shows that the inequality in (b) is also sharp. Finally,
Example 5.10 will show that the $\Bbbk$-linear independence of the supports is
also needed.

The following example shows that schemes that are irredundant to
$F\in\mathcal{S}_{d}$ may be not $d$-regular.

######  Example 5.8.

Let us consider the scheme $Z$ evinced by the GAD
$F=X_{0}G_{1}+X_{1}G_{2}\in\mathcal{S}_{4}$, where

$\displaystyle G_{1}$
$\displaystyle=10X_{0}^{3}-4X_{0}^{2}X_{1}+4X_{0}^{2}X_{2}-4X_{0}X_{1}^{2}-8X_{0}X_{1}X_{2}-3X_{0}X_{2}^{2}-8X_{1}^{3}-4X_{2}^{3}\in\mathcal{S}_{3},$
$\displaystyle G_{2}$
$\displaystyle=5X_{0}^{3}+9X_{0}X_{1}^{2}-5X_{1}^{3}-7X_{1}^{2}X_{2}+6X_{1}X_{2}^{2}-X_{2}^{3}\in\mathcal{S}_{3}.$

Its defining ideal is

$I(Z)=(Y_{0}^{3}Y_{1}^{3}-2Y_{0}^{3}Y_{2}^{3}+5Y_{1}^{3}Y_{2}^{3},3Y_{0}^{2}Y_{1}Y_{2}-2Y_{0}Y_{2}^{3},Y_{0}Y_{1}^{2}Y_{2},Y_{0}Y_{1}Y_{2}^{2},Y_{2}^{4}),$

whose minimal primary decomposition is $I(Z)=I_{1}\cap I_{2}$, where

$I_{1}=(-3Y_{0}Y_{1}Y_{2}+Y_{1}^{3},Y_{1}^{2}Y_{2},Y_{1}Y_{2}^{2},Y_{1}^{3}-2Y_{2}^{3}),\quad
I_{2}=(Y_{2}^{4},Y_{0}^{3}+5Y_{2}^{3},Y_{0}Y_{2}).$

Its Hilbert function is $\mathrm{HF}_{Z}=(1,3,6,10,11,12,12,\dots)$, hence $Z$
is not regular in degree $4=\deg(F)$.

######  Claim.

$Z$ is irredundant to $F$.

###### Proof of Claim..

The connected components of $Z$ are both contained in $4$-fat points, i.e.
$I_{i}\supset\wp_{X_{i-1}}^{4}$, hence by Corollary 4.5 it is sufficient to
show that the unique scheme $Y\subseteq Z$ evinced by a GAD of $F$ of type
$F=X_{0}^{a_{0}}Q_{1}+X_{1}^{a_{1}}Q_{2}$ with $a_{0},a_{1}\geq 1$ is $Z$
itself. Since in the expression of $F$ appear the monomials $-4X_{0}X_{2}^{3}$
and $-X_{1}X_{2}^{3}$, then it is easy to see that there is no such a GAD of
$F$ for $a_{0}>1$ or $a_{1}>1$, therefore we assume $a_{0}=a_{1}=1$.

Since this new additive decomposition is still equal to $F$, we have

$X_{0}(Q_{1}-G_{1})+X_{1}(Q_{2}-G_{2})=0,$

hence there is $T\in\mathcal{S}_{2}$ such that

$X_{1}T=Q_{1}-G_{1},\quad X_{2}T=-Q_{2}+G_{2}.$

This means that $Y$ is evinced by a GAD of $F$ of type

$F=X_{0}(G_{1}+X_{1}T)+X_{1}(G_{2}-X_{0}T),$

for some

$T=\lambda_{1}X_{0}^{2}+\lambda_{2}X_{0}X_{1}+\lambda_{3}X_{0}X_{2}+\lambda_{4}X_{1}^{2}+\lambda_{5}X_{1}X_{2}+\lambda_{6}X_{2}^{2}\in\mathcal{S}_{2}.$

If $Y=Y_{1}\cup Y_{2}\subseteq Z$, then we have

$I_{1}\subseteq
I(Y_{1})=I(Z_{X_{0}(G_{1}+X_{1}T),X_{0}})\subseteq\textnormal{Ann}\big{(}X_{0}(G_{1}+X_{1}T)\big{)},$

which implies

$\begin{cases}0=(-3Y_{0}Y_{1}Y_{2}+Y_{1}^{3})\circ\big{(}X_{0}(G_{1}+X_{1}T)\big{)}=6(-\lambda_{3}+\lambda_{4})X_{0}-6\lambda_{5}X_{1}-6\lambda_{6}X_{2},\\\
0=(Y_{1}^{2}Y_{2})\circ\big{(}X_{0}(G_{1}+X_{1}T)\big{)}=2\lambda_{5}X_{0},\\\
0=(Y_{1}Y_{2}^{2})\circ\big{(}X_{0}(G_{1}+X_{1}T)\big{)}=2\lambda_{6}X_{0},\\\
0=(Y_{1}^{3}-2Y_{2}^{3})\circ\big{(}X_{0}(G_{1}+X_{1}T)\big{)}=6\lambda_{4}X_{0}.\end{cases}$

Similarly, from
$I_{2}\subseteq\textnormal{Ann}\big{(}X_{1}(G_{2}+X_{0}T)\big{)}$ we obtain

$\begin{cases}0=(Y_{2}^{4})\circ\big{(}X_{1}(G_{2}+X_{0}T)\big{)}=0,\\\
0=(Y_{0}^{3}+5Y_{2}^{3})\circ\big{(}X_{1}(G_{2}+X_{0}T)\big{)}=-6\lambda_{1}X_{1},\\\
0=(Y_{0}Y_{2})\circ\big{(}X_{1}(G_{2}+X_{0}T)\big{)}=-2\lambda_{3}X_{0}X_{1}-\lambda_{5}X_{1}^{2}-2\lambda_{6}X_{1}X_{2}.\end{cases}$

The above systems imply

$\lambda_{1}=\lambda_{3}=\lambda_{4}=\lambda_{5}=\lambda_{6}=0,$

thus we conclude that $T=\lambda_{2}X_{0}X_{1}$. We computationally verify
that the scheme evinced by the GAD

$X_{0}(G_{1}+\lambda_{2}X_{0}X_{1}^{2})+X_{1}(G_{2}-\lambda_{2}X_{0}^{2}X_{1})$

is independent on $\lambda_{2}\in\Bbbk$, and it is always equal to $I(Z)$.
Therefore, we conclude that $Y=Z$, i.e. $Z$ is irredundant. ∎

The proof of the above claim shows an effective way for establishing
irredundancy to $F$ by symbolically testing its GADs. $\spadesuit$

###  5.2. Tangential decompositions

In this section, we prove that if a minimal apolar scheme to
$F\in\mathcal{S}_{d}$ is a union of simple points and $2$-jets (i.e. local
$0$-dimensional schemes of length $2$), then it is $d$-regular. Such schemes
are evinced by GADs as in eq. 9, which are called _tangential decompositions_
due to their relation with secant varieties of tangential varieties of
Veronese varieties [BT20, CGG].

######  Proposition 5.9.

Let $Z=Z_{1}\cup\ldots\cup Z_{r}$ such that ${\rm len}(Z_{i})\leq 2$ for every
$i\in\\{1,\dots,r\\}$. If $Z$ is of minimal length among the apolar schemes to
$F\in\mathcal{S}_{d}$, then $Z$ is $d$-regular.

###### Proof.

By Corollary 4.5, $Z$ is evinced by a GAD of $F$ of type

(9) $F=\sum_{i=1}^{s}L_{i}^{d-1}G_{i}+\sum_{i=s+1}^{r}L_{i}^{d}$

for some $0\leq s\leq r$ and $L_{i},G_{i}\in\mathcal{S}_{1}$. Moreover, we
have

$I(Z)^{\perp}_{d}=\langle
L_{i}^{d},L_{j}^{d-1}G_{j}\rangle_{\begin{subarray}{c}1\leq i\leq r\\\ 1\leq
j\leq s\end{subarray}}.$

Since ${\rm len}(Z)$ is $r+s$, which is also equal to the number of generators
of $I(Z)^{\perp}_{d}$, in order to prove that $Z$ is $d$-regular it is
sufficient to show that all those generators are $\Bbbk$-linearly independent.
We prove that if there is a linear relation between the $L_{i}^{d}$’s and the
$L_{j}^{d-1}G_{j}$’s as above, then we can explicitly produce an apolar scheme
that has smaller length than $Z$, contradicting its minimality.

When such a relation involves an addenda appearing in the above GAD of $F$,
then $Z$ is redundant by Proposition 5.2, contradicting the minimality. Thus,
we only need to show that $L_{1}^{d},\ldots,L_{s}^{d}$ are linearly
independent. We will prove a stronger fact, namely that
$L_{1}^{d-1},\ldots,L_{s}^{d-1}$ are linearly independent. Suppose by
contradiction that $L_{1}^{d-1}=\sum_{i=2}^{s}\lambda_{i}L_{i}^{d-1}$ for some
$\lambda_{i}\in\Bbbk$. By substituting this relation in the above GAD, we get

$F=\sum_{i=2}^{s}L_{i}^{d-1}(G_{i}+\lambda_{i}G_{1})+\sum_{i=s+1}^{r}L_{i}^{d}.$

The scheme $Z^{\prime}$ evinced by this new GAD of $F$ has length at most
$s+r-2={\rm len}(Z)-2<{\rm len}(Z)$. ∎

Notice that in the proof of Proposition 5.9 we have employed the length-
minimality of the scheme $Z$ apolar to $F$. Indeed, the irredundancy of an
apolar scheme of $2$-jets is not sufficient to guarantee the regularity in the
degree of $F$, as shown in the following example.

######  Example 5.10.

Let $Z$ be the scheme evinced by the GAD

$F=X_{0}^{2}X_{2}+X_{1}^{2}X_{3}+(X_{0}+X_{1})^{2}X_{4}+(X_{0}-X_{1})^{2}(X_{2}-3X_{3}-2X_{4})+(X_{0}+2X_{1})^{2}(X_{2}+X_{3}+X_{4})\in\mathcal{S}_{3}.$

It is easy to check that $F$ is written in essential variables [IK99, Car06],
and that $Z$ is the union of five $2$-jets $Z_{1},\ldots,Z_{5}$ supported on
points $[L_{1}],\dots,[L_{5}]\in\mathbb{P}^{n}$ of the rational normal cubic.
Its Hilbert function is $\mathrm{HF}_{Z}=(1,5,8,9,10,10,\ldots)$, therefore
$Z$ is not regular in degree $3=\deg(F)$.

However, $Z$ is irredundant: any proper subscheme of $Z=\cup_{i=1}^{5}Z_{i}$
has to be contained in one of the following, for $i\in\\{1,\dots,5\\}$:

$Y_{i}=[L_{i}]\cup\bigcup_{j\neq i}Z_{j}.$

We computationally verify that for every $i$ we have
$I(Y_{i})\subsetneq\textnormal{Ann}(F)$, therefore no proper subscheme of $Z$
is apolar to $F$.

We now verify that the strategy of Proposition 5.9 produces an apolar scheme
that is shorter than $Z$, but not contained in it. Substituting the relation

$(X_{0}-X_{1})^{2}=2X_{0}^{2}+2X_{1}^{2}-(X_{0}+X_{1})^{2}$

we obtain the new GAD of $F$:

$X_{0}^{2}(3X_{2}-6X_{3}-4X_{4})+X_{1}^{2}(2X_{2}-5X_{3}-4X_{4})+(X_{0}+X_{1})^{2}(-X_{2}+3X_{3}+3X_{4})+(X_{0}+2X_{1})^{2}(X_{2}+X_{3}+X_{4}).$

The scheme evinced by this GAD has length $8$ but is not contained in $Z$. We
can repeat the procedure with the relation

$(X_{0}+2X_{1})^{2}=2(X_{0}+X_{1})^{2}-X_{0}^{2}+2X_{1}^{2},$

which leads us to another GAD

$F=X_{0}^{2}(2X_{2}-7X_{3}-5X_{4})+X_{1}^{2}(4X_{2}-3X_{3}-2X_{4})+(X_{0}+X_{1})^{2}(X_{2}+5X_{3}+5X_{4}).$

The scheme evinced the last GAD is minimal among the apolar schemes to $F$: as
it has length $6$ and, up to a change of variables, $F$ is the Perazzo cubic
[Per00] which has cactus rank 6 (see eg. [BBM14, Example 2.8], [BB15, Section
4]). This can also be directly verified with [BT20, Algorithm 3]. $\spadesuit$

###  5.3. Apolar schemes with low length

######  Proposition 5.11.

Let $Z\subset\mathbb{P}^{n}$ be a $0$-dimensional scheme apolar and
irredundant to $F\in\mathcal{S}_{d}$. If ${\rm len}(Z)\leq 2d+1$, then $Z$ is
$d$-regular.

###### Proof.

By contradiction, let us assume that $Z$ is not $d$-regular. Then, by [BGI11,
Lemma 34], there exists a line $L$ such that ${\rm len}(Z\cap L)\geq d+2$. Let
${\rm Res}_{L}(Z)$ be the residual scheme of $Z$ with respect to $L$ defined
by the colon ideal $\big{(}I(Z):(L)\big{)}$. Since

${\rm len}(Z\cap L)+{\rm len}\big{(}{\rm Res}_{L}(Z)\big{)}={\rm len}(Z)\leq
2d+1,$

then given the irreducible decomposition $Z=Z_{1}+\cdots+Z_{s}$, there exists
a component $Z_{i}$ such that the schematic intersection $Z_{i}\cap L$
satisfies ${\rm len}(Z_{i}\cap L)>{\rm len}({\rm Res}_{L}(Z_{i}))$. Without
loss of generality, we may assume that $i=1$, $I(Z_{1})\subseteq\wp_{X_{0}}$
and $I(L)=(X_{1},\dots,X_{n})$. Let $H$ be the orthogonal hyperplane to
$X_{0}$, i.e. $I(H)=(X_{0})$, and let $m={\rm len}(Z_{1}\cap L)$. We consider
the scheme $Z^{\prime}$ defined by

$I(Z^{\prime})=I\left(Z_{1}\cap(m-1)H\right)\cap I(Z_{2})\cap\dots\cap
I(Z_{s}).$

It is clear that $Z^{\prime}\subsetneq Z$, hence to get the desired
contradiction it is sufficient to show that $Z^{\prime}$ is apolar to $F$,
which follows directly from the following fact by the Apolarity Lemma (Lemma
2.3).

######  Claim 5.1.

$I(Z)^{\perp}_{d}=I(Z^{\prime})^{\perp}_{d}$.

_Proof of 5.1._

Since $m>{\rm len}\big{(}{\rm Res}(Z_{1})\big{)}$ we have
$(X_{0}^{m-1})\cap(X_{1},\ldots,X_{n})\subseteq I(Z_{1})$, hence

$I(Z_{1})=\big{(}I(Z_{1})+(X_{0}^{m-1})\big{)}\cap\big{(}I(Z_{1})+(X_{1},\ldots,X_{m})\big{)}.$

$\bullet$ We prove that $I(Z_{1})+(X_{0}^{m-1})$ equals the saturated ideal
$I\big{(}Z_{1}\cap(m-1)H\big{)}$.

There are obvious ideal inclusions:

(10) $I(Z_{1})\subseteq I(Z_{1})+(X_{0}^{m-1})\subseteq
I\big{(}Z_{1}\cap(m-1)H\big{)}.$

It is enough to show that the last two ideals have the same Hilbert function.
Since $Z_{1}\cap(m-1)H$ has colength $1$ inside $Z_{1}$ and their homogeneous
defining ideals agree up to degree $m-2$, we deduce that

$\mathrm{HF}_{Z_{1}\cap(m-1)H}(i)=\begin{cases}\mathrm{HF}_{Z_{1}}(i)&\text{for
$i\leq m-2$,}\\\ \mathrm{HF}_{Z_{1}}(i)-1&\text{for $i\geq m-1$.}\end{cases}$

By eq. 10 the Hilbert function $\mathrm{HF}_{*}$ of
$\mathcal{S}/\big{(}I(Z_{1})+(X_{0}^{m-1})\big{)}$ is squeezed:
$\mathrm{HF}_{Z_{1}\cap(m-1)H}\leq\mathrm{HF}_{*}\leq\mathrm{HF}_{Z_{1}}$.
However, for every $k\geq m-1$ we have
$X_{0}^{k}\in\left(I(Z_{1})+(X_{0}^{m-1})\right)\setminus I(Z_{1})$, thus
$\mathrm{HF}_{*}(k)<\mathrm{HF}_{Z_{1}}(k)$ for every $k\geq m-1$. This
implies that $\mathrm{HF}_{*}$ completely agrees with
$\mathrm{HF}_{Z_{1}\cap(m-1)H}$.

$\bullet$ For every $i\in\\{2,\ldots,s\\}$, we trivially have

$I(Z_{i})=I(Z_{i})\cap\big{(}I(Z_{i})+(X_{1},\ldots,X_{n})\big{)}.$

Hence, we can write:

$\displaystyle I(Z)$
$\displaystyle=I\big{(}Z_{1}\cap(m-1)H\big{)}\cap\big{(}I(Z_{1})+(X_{1},\ldots,X_{m})\big{)}\cap\bigcap_{i=2}^{s}\big{(}I(Z_{i})+(X_{1},\ldots,X_{n})\big{)}\cap
I(Z_{i})$
$\displaystyle=I(Z^{\prime})\cap\left(\bigcap_{i=1}^{s}I(Z_{i})+(X_{1},\ldots,X_{n})\right)=I(Z^{\prime})\cap
I(Z\cap L).$

$\bullet$ From the non-degeneracy of the apolar action we get

$I(Z)_{d}^{\perp}=[I(Z^{\prime})\cap I(Z\cap
L)]_{d}^{\perp}=I(Z^{\prime})_{d}^{\perp}+I(Z\cap L)_{d}^{\perp}$

but $I(Z\cap L)_{d}=I(Z^{\prime}\cap L)_{d}$ because they define schemes of
length $d+1$ on the same normal curve $\nu_{d}(L)\subset\mathbb{P}^{d}$. Thus,
we conclude

$I(Z)_{d}^{\perp}=I(Z^{\prime})_{d}^{\perp}+I(Z^{\prime}\cap
L)_{d}^{\perp}=I(Z^{\prime})_{d}^{\perp},$

which proves the claim and then concludes the proof. ∎

We notice that Proposition 5.11 provides a good criterion for proving that the
minimal apolar schemes to a _given_ $F\in\mathcal{S}_{d}$ is $d$-regular, by
exhibiting at least one scheme $Z$ apolar to $F$ and of length not bigger than
$2d+1$.

######  Example 5.12.

Let $F\in\mathcal{S}_{4}$ be the polynomial considered in Example 5.8. We
consider another GAD $F=X_{0}\tilde{G}_{1}+X_{1}\tilde{G}_{2}$, where

$\displaystyle\tilde{G}_{1}$
$\displaystyle=10X_{0}^{3}+X_{0}^{2}X_{1}+4X_{0}^{2}X_{2}-4X_{0}X_{1}^{2}-8X_{0}X_{1}X_{2}-3X_{0}X_{2}^{2}-4X_{2}^{3}\in\mathcal{S}_{3},$
$\displaystyle\tilde{G}_{2}$
$\displaystyle=X_{0}X_{1}^{2}-5X_{1}^{3}-7X_{1}^{2}X_{2}+6X_{1}X_{2}^{2}-X_{2}^{3}\in\mathcal{S}_{3}.$

This GAD evinces the scheme $\tilde{Z}$ defined by

$I(\tilde{Z})=\left(Y_{0}^{2}Y_{1}Y_{2}-\frac{2}{3}Y_{0}Y_{2}^{3},Y_{0}Y_{1}Y_{2}^{2},Y_{2}^{4},Y_{0}Y_{1}^{2}-\frac{5}{2}Y_{0}Y_{1}Y_{2}+Y_{2}^{3}\right).$

Its Hilbert function is $\mathrm{HF}_{Z}=(1,3,6,9,9,\dots)$. Since ${\rm
len}(Z)=9\leq 2\cdot 4+1$, by Proposition 5.11 we can guarantee that minimal
schemes apolar to such a $F$ are $4$-regular, even without computing them.

##  6\. Conclusion

In the present work, we investigated the $d$-regularity of certain families of
schemes apolar to $F\in\mathcal{S}_{d}$. In all the examples we presented, the
schemes of minimal lengths were $d$-regular, so it is natural to ask whether
this is always the case.

######  Question 1.

Let $F\in\mathcal{S}_{d}$ and $Z$ be a $0$-dimensional scheme evincing its
cactus rank. Is $Z$ $d$-regular?

Actually, a careful reader would have noticed that none of the examples we
considered really required to reach degree $d$ for regularity, hence we may
state an even more compelling question.

######  Question 2.

Let $F\in\mathcal{S}_{d}$ and $Z$ be a $0$-dimensional scheme evincing its
cactus rank. Is $Z$ $(d-1)$-regular?

To the best of our knowledge, we do not know the answer to 1 and 2. We believe
that our results and examples could be useful in either direction. Our
positive results restrict the identikit of a possible example providing a
negative answer to 1 to have some component of high multiplicity. On the other
side, when trying to prove a positive answer to 1 or 2, Example 4.4 shows that
we really need to use the global assumption of minimality in terms of the
cactus rank, and that cannot be relaxed with the local condition of minimality
by inclusion.

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