Stability of associated forms

Fedorchuk, Maksym; Isaev, Alexander

doi:10.1090/jag/719

Cited by 3 publications

(11 citation statements)

References 25 publications

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“…Suppose

x_{1}^{d_{1}} \dots x_{n}^{d_{n}}

is the smallest with respect to the graded reverse lexicographic order monomial of degree

n (d - 1)

that does not lie in

{false(J_{f} false)}_{n false(d - 1 false)}

. Since

z_{1}^{d_{1}} \dots z_{n}^{d_{n}}

must appear with a non‐zero coefficient in

A (f)

, we have that

d_{1} + \dots + d_{a} = deg G_{1} .

On the other hand, by [, Lemma 4.1], we have that

d_{1} + \dots + d_{a} ⩽ a false(d - 1 false)

. It follows that

deg G_{1} ⩽ a false(d - 1 false)

.…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…Consider a balanced direct sum

U = false⟨ g_{1}, \dots, g_{n} false⟩ \in Grass {(n, k {[x_{1}, \dots, x_{n}]}_{d})}_{prefixRes}

, where

g_{1}, \dots, g_{a} \in k {[x_{1}, \dots, x_{a}]}_{d}

and

g_{a + 1}, \dots, g_{n} \in k {[x_{a + 1}, \dots, x_{n}]}_{d}

. Then, up to a non‐zero scalar,

A false(U false) = A false(g_{1}, \dots, g_{a} false) A false(g_{a + 1}, \dots, g_{n} false),

where

A false(g_{1}, \dots, g_{a} false) \in k {[z_{1}, \dots, z_{a}]}_{a (d - 1)}

and

A false(g_{a + 1}, \dots, g_{n} false) \in k {[z_{a + 1}, \dots, z_{n}]}_{(n - a) (d - 1)}

; see [, Lemma 2.11], which also follows from the fact that on the level of algebras, we have

\frac{k [x_{1}, \dots, x_{n}]}{(g_{1}, \dots, g_{n}}

…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…It follows that the top degree of

R

is strictly less than

a (d - 1)

, and so

I_{a (d - 1)}^{'} = k {[x_{1}, \dots, x_{a}]}_{a (d - 1)}

(cf. [, Lemma 2.7]). But then

k {[x_{1}, \dots, x_{a}]}_{a (d - 1)} \subset false(x_{a + 1}, \dots, x_{n} false) + I_{U} .

Using , this gives

k {[x_{1}, \dots, x_{a}]}_{a (d - 1)} k {[x_{a + 1}, \dots, x_{n}]}_{(n - a) (d - 1)} \subset I_{U} .

Thus, every monomial of

k {[z_{1}, \dots, z_{a}]}_{a (d - 1)} k {[z_{a + 1}, \dots, z_{n}]}_{(n - a) (d - 1)}

appears with coefficient

0

A (U)

, which contradicts .

□

…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…A homogeneous polynomial

f

is called a direct sum if, after a change of variables, it can be written as a sum of two or more polynomials in disjoint sets of variables:

f = f_{1} false(x_{1}, \dots, x_{a} false) + f_{2} false(x_{a + 1}, \dots, x_{n} false) .

Homogeneous direct sums are the subject of a well‐known symmetric Strassen's additivity conjecture postulating that the Waring rank of

f

in is the sum of the Waring ranks of

f_{1}

and

f_{2}

(see, for example, ). Direct sums also play a special role in the study of geometric invariant theory (GIT) stability of associated forms .…”

Section: Introductionmentioning

confidence: 99%

“…Recall that to a smooth form

f

of degree

d + 1

n

variables, one can assign a degree

n (d - 1)

form

A (f)

n

(dual) variables, called the associated form of

f

. The associated form

A (f)

is defined as a Macaulay inverse system of the Milnor algebra of

f

, which simply means that the apolar ideal of

A (f)

coincides with the Jacobian ideal of

f :

A {(f)}^{⊥} = false(\partial f / \partial x_{1}, \dots, \partial f / \partial x_{n} false) .

Such definition leads to an observation that for a smooth form

f

that is written as a sum of two forms in disjoint sets of variables, the associated form

A (f)

decomposes as a product of two forms in disjoint sets of (dual) variables ([, Lemma 2.11]). For example, up to a scalar,

A false(x_{1}^{d + 1} + \dots + x_{n}^{d + 1} false) = z_{1}^{d - 1} \dots z_{n}^{d - 1} .

The main purpose of Theorem is to prove the converse statement, and thus establish an if‐and‐only‐if criterion for direct sum decomposability of a smooth form

f

in terms of the factorization properties of its associated form

A (f)

(see Theorem ).…”

Section: Introductionmentioning

confidence: 99%

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Direct sum decomposability of polynomials and factorization of associated forms

Fedorchuk

2020

Proc. London Math. Soc.

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We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non‐zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if‐and‐only‐if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.

show abstract

“…Suppose

x_{1}^{d_{1}} \dots x_{n}^{d_{n}}

is the smallest with respect to the graded reverse lexicographic order monomial of degree

n (d - 1)

that does not lie in

{false(J_{f} false)}_{n false(d - 1 false)}

. Since

z_{1}^{d_{1}} \dots z_{n}^{d_{n}}

must appear with a non‐zero coefficient in

A (f)

, we have that

d_{1} + \dots + d_{a} = deg G_{1} .

On the other hand, by [, Lemma 4.1], we have that

d_{1} + \dots + d_{a} ⩽ a false(d - 1 false)

. It follows that

deg G_{1} ⩽ a false(d - 1 false)

.…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…Consider a balanced direct sum

U = false⟨ g_{1}, \dots, g_{n} false⟩ \in Grass {(n, k {[x_{1}, \dots, x_{n}]}_{d})}_{prefixRes}

, where

g_{1}, \dots, g_{a} \in k {[x_{1}, \dots, x_{a}]}_{d}

and

g_{a + 1}, \dots, g_{n} \in k {[x_{a + 1}, \dots, x_{n}]}_{d}

. Then, up to a non‐zero scalar,

A false(U false) = A false(g_{1}, \dots, g_{a} false) A false(g_{a + 1}, \dots, g_{n} false),

where

A false(g_{1}, \dots, g_{a} false) \in k {[z_{1}, \dots, z_{a}]}_{a (d - 1)}

and

A false(g_{a + 1}, \dots, g_{n} false) \in k {[z_{a + 1}, \dots, z_{n}]}_{(n - a) (d - 1)}

; see [, Lemma 2.11], which also follows from the fact that on the level of algebras, we have

\frac{k [x_{1}, \dots, x_{n}]}{(g_{1}, \dots, g_{n}}

…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…It follows that the top degree of

R

is strictly less than

a (d - 1)

, and so

I_{a (d - 1)}^{'} = k {[x_{1}, \dots, x_{a}]}_{a (d - 1)}

(cf. [, Lemma 2.7]). But then

k {[x_{1}, \dots, x_{a}]}_{a (d - 1)} \subset false(x_{a + 1}, \dots, x_{n} false) + I_{U} .

Using , this gives

k {[x_{1}, \dots, x_{a}]}_{a (d - 1)} k {[x_{a + 1}, \dots, x_{n}]}_{(n - a) (d - 1)} \subset I_{U} .

Thus, every monomial of

k {[z_{1}, \dots, z_{a}]}_{a (d - 1)} k {[z_{a + 1}, \dots, z_{n}]}_{(n - a) (d - 1)}

appears with coefficient

0

A (U)

, which contradicts .

□

…”

Section: Direct Sum Decomposability Of Smooth Formsmentioning

confidence: 99%

“…A homogeneous polynomial

f

is called a direct sum if, after a change of variables, it can be written as a sum of two or more polynomials in disjoint sets of variables:

f = f_{1} false(x_{1}, \dots, x_{a} false) + f_{2} false(x_{a + 1}, \dots, x_{n} false) .

Homogeneous direct sums are the subject of a well‐known symmetric Strassen's additivity conjecture postulating that the Waring rank of

f

in is the sum of the Waring ranks of

f_{1}

and

f_{2}

(see, for example, ). Direct sums also play a special role in the study of geometric invariant theory (GIT) stability of associated forms .…”

Section: Introductionmentioning

confidence: 99%

“…Recall that to a smooth form

f

of degree

d + 1

n

variables, one can assign a degree

n (d - 1)

form

A (f)

n

(dual) variables, called the associated form of

f

. The associated form

A (f)

is defined as a Macaulay inverse system of the Milnor algebra of

f

, which simply means that the apolar ideal of

A (f)

coincides with the Jacobian ideal of

f :

A {(f)}^{⊥} = false(\partial f / \partial x_{1}, \dots, \partial f / \partial x_{n} false) .

Such definition leads to an observation that for a smooth form

f

that is written as a sum of two forms in disjoint sets of variables, the associated form

A (f)

decomposes as a product of two forms in disjoint sets of (dual) variables ([, Lemma 2.11]). For example, up to a scalar,

A false(x_{1}^{d + 1} + \dots + x_{n}^{d + 1} false) = z_{1}^{d - 1} \dots z_{n}^{d - 1} .

The main purpose of Theorem is to prove the converse statement, and thus establish an if‐and‐only‐if criterion for direct sum decomposability of a smooth form

f

in terms of the factorization properties of its associated form

A (f)

(see Theorem ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Direct sum decomposability of polynomials and factorization of associated forms

Fedorchuk

2020

Proc. London Math. Soc.

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Divisors on the Moduli Space of Curves From Divisorial Conditions On Hypersurfaces

Tseng

2021

Experimental Mathematics

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Associated Form Morphism

Fedorchuk

Isaev

2019

International Mathematics Research Notices

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We study the geometry of the morphism that sends a smooth hypersurface of degree d + 1 in P n−1 to its associated hypersurface of degree n(d − 1) in the dual space P n−1 ∨ .given by a categorical quotient of the locus of GIT semistable hypersurfaces. We call V m,n the GIT compactification of U m,n . The subject of this paper is a certain rational map V m,n V n(m−2),n , where n ≥ 2, m ≥ 3 and where we exclude the (trivial) case (n, m) = (2, 3). While this map has a purely algebraic construction, which we shall recall soon, it has several surprising geometric properties that we establish in this paper. In particular, this rational map restricts to a locally closed immersionĀ : U m,n → V n(m−2),n , and often Mathematics Subject Classification: 14L24, 13A50, 13H10.

show abstract

Stability of associated forms

Abstract: We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type (d, . . . , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.

Cited by 3 publications

References 25 publications

Direct sum decomposability of polynomials and factorization of associated forms

Direct sum decomposability of polynomials and factorization of associated forms

Divisors on the Moduli Space of Curves From Divisorial Conditions On Hypersurfaces

Associated Form Morphism

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