Download Three Directions In Waring Ranks and Apolarity

Three Directions In Waring Ranks and Apolarity
Zach Teitler
Boise State University
Question
Definition
The Waring rank of F is the least r such that F = `d1 + · · · + `dr for some linear forms `i .
When is f (X , Y ) = g (X ) + h(Y )?
Example
1
1
2
xy = (x + y ) − (x − y )2
4
4
1
xyz =
(x + y + z)3 − (x + y − z)3 − (x − y + z)3 + (x − y − z)3
24
Indeed r (xy ) = 2 and r (xyz) = 4.
Known values of rank:
1 Quadratic forms
2 Binary forms
3 General forms (Alexander-Hirschowitz, 1995)
4 Monomials and sums of coprime monomials (Carlini-Catalisano-Geramita, 2011)
Definition
Example
Waring rank is X = νd (Pn ) (Veronese)
Tensor rank is X = Seg(Pn1 × · · · × Pnk ) (Segre)
Alternating tensor rank is X = G (k, `) (Grassmanian)
We allow linear changes of coordinates.
Example
Certainly xy 6= g (x) + h(y ) but
1
1
2
xy = (x + y ) − (x − y )2.
4
4
Example
Let F be a multihomogeneous form of multidegree (d1, . . . , dn ). The multihomogeneous rank is the least
d
d
r such that F = (`111 · · · `d1nn ) + · · · + (`r 11 · · · `drnn ). This is rank with respect to a Segre-Veronese variety.
Question (M. Sepideh Shafiei)
Example
Let W ⊂ P(S d V ) be a linear series. The simultaneous
Waring rank is the least r such that there
P d
exist some linear forms `1, . . . , `r so that f =
ci `i for every f ∈ W .
Let detn = det((xij )1≤i,j≤n ). We have
x11
det2 = det
x21
Question
Let X ⊂ Pn and q ∈ Pn . The rank of q with respect to X is the least r = rX (q) such that
q ∈ span(x1, . . . , xr ) for distinct, reduced points xi ∈ X .
x12
x22
= x11x22 − x12x21.
Is detn decomposable in this way for n > 2?
What is r (F ) if F is a product of linear forms, possibly with repetition
(defining equation of a hyperplane multiarrangement)?
In joint work with Alex Woo (University of Idaho) we address this when the hyperplane
arrangement arises as the set of mirrors of a finite reflection group.
Example
Q
The classical Vandermonde determinant Vn = i<j (xi − xj ) defines the set of mirrors of the symmetric
group Sn . We show that r (Vn ) = (n − 1)!.
For the upper bound:
n
1 X
(
Vn =
(sgn π)(π`) 2)
|Sn |
It is easy to see that detn is indecomposable in the original variables, but not easy to determine
decomposability when linear changes of coordinates are allowed.
Definition
Let f ∈ S d V . A direct sum decomposition is f = f1 + f2, 0 6= fi ∈ S d Vi , V1 ⊕ V2 = V .
In joint work with Weronika Buczyńska (IMPAN) and Jaroslaw Buczyński (IMPAN) we give a
necessary condition for decomposability in terms of apolarity.
A point of rank 3.
Theorem (Buczyńska–Buczyński–T.)
π∈Sn
gives r (Vn ) ≤ |Sn | = n!. We improve this by taking
X
`=
e 2πij/n xj ,
an eigenvector of c = (1 2 · · · n). Terms in the sum are the same in each coset of C = hci. So we
reduce to a sum over cosets of C :
X
n
1
(
(sgn σ)(σ`) 2)
Vn =
|Sn /C |
σ∈Sn /C
showing r (Vn ) ≤ |Sn /C | = (n − 1)!.
For the lower bound we recall well-known invariant theory: the apolar algebra AVn , the vector space
spanned by all the derivatives of Vn of all orders, has dimension |Sn |; and the apolar annihilating
ideal Vn⊥ is generated exactly in the degrees of Sn , namely 2, 3, . . . , n.
By the theorem of Ranestad-Schreyer (2011), r (Vn ) ≥ dim(AVn )/δ, where δ is the generating degree of
Vn⊥. So r (Vn ) ≥ n!/n = (n − 1)!.
If f is decomposable then the apolar ideal f ⊥ has a minimal generator in degree d = deg f .
Example
Shafiei has shown that det⊥
n is generated in degree 2. Therefore detn is indecomposable for n > 2.
The converse does not hold:
Example
xy d −1 is indecomposable (because x d − y d factors with distinct factors) yet (xy d −1)⊥ = hX 2, Y d i has
a generator of degree d .
Note that
d − yd
(y
+
tx)
xy d −1 = lim
dt
t→0
Let L be a very ample line bundle on X , V = PH 0(X , L)∗, and E a vector bundle on X of rank (fiber
dimension) e. Let q ∈ V . From the multiplication map of bundles
(L ⊗ E ∗) ⊗ E → L
we get a multiplication map of global sections
the last map being given by the fixed v ∈ V .
The (generalized) catalecticant of v with respect to E is
is a limit of direct sums.
CvE : H 0(L ⊗ E ∗) → H 0(E )∗.
Theorem (Buczyńska–Buczyński–T.)
A reflection is a diagonalizable map on Cn that fixes a hyperplane.
It is a real reflection if it has order 2.
A reflection group is a group generated by reflections.
Example
Sylvester’s catalecticant corresponds to X = νd (Pn ), L = O(d ), and E = O(a).
Proposition (Landsberg–Ottaviani)
(y (x 2 + yz))⊥ has two cubic generators;
The fundamental skew invariant fG of a finite reflection group G is the defining equation of the
arrangement of mirror hyperplanes of reflections in G , each hyperplane with multiplicity k − 1 when it
is the mirror of a reflection of order k.
1 2 3
(y
+
tx
+
2t
z)
2
6t
⊕(y − tx)3 ⊕ −2y 3
In the natural action of G on polynomials, a form f is a skew invariant if gf = (det g )−1f for all
g ∈ G . fG is a skew invariant and divides every skew invariant.
−→ y (x 2 + yz)
Theorem (T.–Woo)
Let G be a finite reflection group whose largest degree D is a regular number (this includes all real
|G |
reflection groups). Then r (fG ) =
.
D
Definition
H 0(L ⊗ E ∗) ⊗ H 0(E ) → H 0(L) = V ∗ → C,
Suppose f ∈ C[x1, . . . , xn ] can not be written using fewer than n variables.
Then f ⊥ has a minimal generator in degree d if and only if f is a limit of direct sums.
Definition
Landsberg-Ottaviani gave a lower bound for rank in terms of generalized catalecticants.
Their lower bound is actually a lower bound for border rank, which can be strictly less than rank.
We give an improvement, to a quantity which is actually a lower bound for rank itself.
This is non-trivial: a limit ft → f does not imply ft⊥ → f ⊥.
Example
(x d − ty d )⊥ 6→ (x d )⊥.
We also consider low-degree generators of f ⊥.
Theorem (Buczyńska–Buczyński–T.)
If d = deg f , f ∈ C[x1, . . . , xn ] can not be written using fewer than n variables, and f ⊥ is generated in
degree δ, then d ≤ (δ − 1)n.
In particular if f ⊥ is generated by quadrics then d ≤ n.
Example
(x1 · · · xn )⊥ = hX12, . . . , Xn2i is generated by quadrics.
(detn )⊥ is generated by quadrics.
If [v ] = x ∈ X then rk(CvE ) ≤ e,
with equality if and only if both L ⊗ E ∗ and E are globally generated at x.
rk(CvE )
For v ∈ V , the rank and border rank satisfy rX (v ) ≥ brX (v ) ≥
.
e
Theorem (T.)
Suppose G is a vector bundle on X and E = L ⊗ (G ∗)b ; more generally assume there is a bundle map
G b → L ⊗ E ∗ whose kernel has no global sections. Assume CvG is onto. Let Σ be the subvariety of
νb
0
PH (X , G ) ,→ PH 0(X , L ⊗ E ∗) defined by the equations in the image of the transpose
(CvE )t : H 0(X , E ) → H 0(X , L ⊗ E ∗)∗.
Then e · rX (v ) > rk CvE + dim Σ.
Linear equations on PH 0(X , L ⊗ E ∗) induce degree b equations on the Veronese image νb (PH 0(X , G )).
Σ is the locus defined by the equations arising from img(CvE )t .
Example
If f is a multihomogeneous polynomial which is concise in a strong sense (strengthening the notion that
f can not be written using fewer variables) then the well known catalecticant lower bound for r (f ) is
improved by the dimension of a singular locus of f .
Example
If W is a linear series then the catalecticant lower bound for the simultaneous Waring rank of W is
improved by the dimension of a simultaneous singular locus of W (the intersection of the singular loci of
all f ∈ W ), i.e., a singular locus of the variety defined by W .
A4 arrangement and polar polyhedron.
http://math.boisestate.edu/~zteitler
[email protected]