Silvia M.C. Pagani, Silvia...
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Power sum polynomials and discrete tomography
Silvia M.C. Pagani, Silvia Pianta
Università Cattolica del Sacro Cuore, Brescia, Italy
Finite Geometry & Friends
Vrije Universiteit Brussel, June 19, 2019
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Outline
1 Motivation
2 Discrete tomography
3 Results
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Motivation
DefinitionLet P = (p1, . . . ,pn+1) be a point of PG(n,q). The correspondingRédei factor is the linear polynomial P · X = p1X1 + . . .+ pn+1Xn+1.
The zeros of P · X are (the Plücker coordinates of) the hyperplanesthrough P.
DefinitionLet S = {Pi : i = 1, . . . , |S|} ⊆ PG(n,q) be a point set. The Rédeipolynomial of S is defined as
RS(X1, . . . ,Xn) :=
|S|∏i=1
Pi · X.
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Motivation
DefinitionLet S = {Pi : i = 1, . . . , |S|} ⊆ PG(n,q) be a point set. The powersum polynomial of S is
GS(X1, . . . ,Xn+1) :=
|S|∑i=1
(Pi · X)q−1.
P. Sziklai wrote:
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Motivation
DefinitionLet S = {Pi : i = 1, . . . , |S|} ⊆ PG(n,q) be a point set. The powersum polynomial of S is
GS(X1, . . . ,Xn+1) :=
|S|∑i=1
(Pi · X)q−1.
P. Sziklai wrote:
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = {(0,0,1)}
GS(X ,Y ,Z ) = Z
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = {(0,1,0), (0,1,1)}
GS(X ,Y ,Z ) = (Y ) + (Y + Z )
= Z
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = {(1,0,0), (0,1,0),(1,1,1)}
GS(X ,Y ,Z ) = Z
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = {(0,0,1), (1,0,0),(0,1,0), (1,1,0)}
GS(X ,Y ,Z ) = Z
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = {(0,0,1), (0,1,0),(0,1,1), (1,1,0), (1,1,1)}
GS(X ,Y ,Z ) = Z
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Example
In PG(2,2)
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
S = PG(2,2)\{(0,0,1)}
GS(X ,Y ,Z ) = Z
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Discrete tomography
Tomography is concerned with the reconstruction of the internal of anobject from the knowledge of its projections taken along givendirections.
In discrete tomography the object (image) is a set of pixels anddirections have rational slope.
Usual assumption: the image is confined in a given finite grid.
3 4 -2 1
0 -1 3 5
0 2 7 3
1
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Uniqueness of reconstruction
One of the main tasks of tomography is to ensure that thereconstructed image equals the original one.
In general, it is not achievable.
2 2
3
1
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Uniqueness of reconstruction
One of the main tasks of tomography is to ensure that thereconstructed image equals the original one.
In general, it is not achievable.
2 2
3
1
0 32 -1
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Uniqueness of reconstruction
One of the main tasks of tomography is to ensure that thereconstructed image equals the original one.
In general, it is not achievable.
2 2
3
1
1 21 0
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Uniqueness of reconstruction
One of the main tasks of tomography is to ensure that thereconstructed image equals the original one.
In general, it is not achievable.
2
+ α0 32 -1 -1
11-1
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Switching components and ghosts
DefinitionA switching component w.r.t. a set U of directions is a pair of sets ofpixels, having the same projections along the directions in U. Theunion of the elements of a switching component (suitably weighted)constitute the support of a ghost, which is a nonzero image with nullprojections along U.
Example: a switching component (left) and a ghost (right) w.r.t. thecoordinate directions.
−1 −1
−1
−1
−1
1 1
1
1
1
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0
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Looking for connections & master plan
Discrete tomography Projective geometry
lattice grid ←→ PG(2,q)
image ←→ S
projections along U ←→ GS
ghost w.r.t. U ←→ S such that GS ≡ 0
How to compute the zeros of GS: look at the intersection of S withlines.
In fact, if |S ∩ `| = m for a line `, then GS(`) = |S| −m.
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Looking for connections & master plan
Discrete tomography Projective geometry
lattice grid ←→ PG(2,q)
image ←→ S
projections along U ←→ GS
ghost w.r.t. U ←→ S such that GS ≡ 0
How to compute the zeros of GS: look at the intersection of S withlines.
In fact, if |S ∩ `| = m for a line `, then GS(`) = |S| −m.
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Some general results
Let q = ph, p prime.
LemmaLet S ⊆ PG(2,q) be a ghost. Then PG(2,q)\S is a ghost.
ProofFor every line `, GS(`) = |S| −m = 0 mod p. Then
GPG(2,q)\S(`) = |PG(2,q)\S| − |(PG(2,q)\S) ∩ `|= q2 + q + 1− |S| − (q + 1−m)
= m − |S| = 0 mod p.
=⇒ ∅ and PG(2,q) are ghosts.
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Some general results
LemmaA partial pencil P of λp + 1 lines, λ = 0, . . . ,ph−1, is a ghost.Consequently, a set of q − λp lines through a point P minus P is aghost.
In particular, it results that a line is a ghost, as well as every affineplane contained in PG(2,q).
ProofEvery line ` meets P in either one, λp + 1 or q + 1 points. In all casesm = 1 mod p and
GP(`) = (λp + 1)q + 1−m = 0 mod p.
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Blocking sets
DefinitionA blocking set (for the lines) of PG(2,q) is a set of points meetingevery line of PG(2,q) and not containing a line.
DefinitionA Baer subplane of PG(2,q), q square, is a subplane of order
√q.
A Baer subplane is a minimal blocking set for lines (with minimum size)and a ghost (each line intersects it in either 1 or
√q + 1 points).
DefinitionA unital of PG(2,q), q square, is a set of q
√q + 1 points meeting every
line of PG(2,q) in either 1 or√
q + 1 points.
A unital is a minimal blocking set with maximum cardinality and aghost.
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Cardinalities
To summarize, there are ghosts with size:
0, q + 1, pq, (p + 1)q + 1, 2pq,
. . . , q2 − (p − 1)q + 1, q2, q2 + q + 1.
Moreover, if q is a square, there exist ghosts with size:
q +√
q + 1, q√
q + 1, q2 − (√
q − 1)q, q2 −√q.
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Operation between sets
In discrete tomography, a ghost can be added to an image.How to move among sets with the same power sum polynomial?What is the corresponding operation in the power sum polynomialcase?
Union seen as multiset (each point is counted modulo thecharacteristic).
GS1∪S2 = GS1 + GS2
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
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Operation between sets
In discrete tomography, a ghost can be added to an image.How to move among sets with the same power sum polynomial?What is the corresponding operation in the power sum polynomialcase? Union seen as multiset (each point is counted modulo thecharacteristic).
GS1∪S2 = GS1 + GS2
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
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Operation between sets
In discrete tomography, a ghost can be added to an image.How to move among sets with the same power sum polynomial?What is the corresponding operation in the power sum polynomialcase? Union seen as multiset (each point is counted modulo thecharacteristic).
GS1∪S2 = GS1 + GS2
(1, 0, 0)
(0, 1, 0)
(1, 1, 0)
(0, 0, 1)
(1, 0, 1)
(0, 1, 1)
(1, 1, 1)
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Conclusions and further work
How to go on:- further investigate the connections between discrete tomography
and power sum polynomials;- find a “basis” for the ghosts (lines?);- (dis)prove that also blocking sets can be seen as (multi)union of
lines;- find other kinds of ghosts, which are neither blocking sets nor
partial pencils.
Thank you for your attention!
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Conclusions and further work
How to go on:- further investigate the connections between discrete tomography
and power sum polynomials;- find a “basis” for the ghosts (lines?);- (dis)prove that also blocking sets can be seen as (multi)union of
lines;- find other kinds of ghosts, which are neither blocking sets nor
partial pencils.
Thank you for your attention!
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