Two Universality Theorems

ShmuelOnn

Technion –Israel Institute of Technology

http://ie.technion.ac.il/~onn

Supported in part by ISF –Israel Science Foundation

Part 1:

GröbnerPolyhedra,Hilbert Polyhedra

and Universal GröbnerBases

Part 2:

MultiwayTables, Markov Bases

and ToricIdeals

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Part 1:

GröbnerPolyhedra, Hilbert Polyhedra

and Universal GröbnerBases

Based on joint papers with Sturmfelsand with Babson& Thomas

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GröbnerPolyhedra

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GröbnerPolyhedra

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Our Goal: Characterizationand efficient computation of

Gröbnerpolyhedraof classesofideals simultaneously.

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GröbnerPolyhedraof Point Configurations

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GröbnerPolyhedraof Point Configurations

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Hilbert Polyhedra

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To handle Gröbnerpolyhedraof classes of ideals

simultaneously, we need the following universal object:

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Hilbert Polyhedra

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The Hilbert Scheme

Some examples of ideals on the Hilbert scheme are:

Unfortunately, this is a very complex object and may be intractable.

But, we next show how to approximate it efficiently and satisfactorily.

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The Hilbert Zonotope

Recall:

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Corollary:

Fix any d. The following hold for the Hilbert scheme:

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Some Problems:

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Part 2:

MultiwayTables, Markov Bases

and ToricIdeals

Based on several joint papers with Jesus De Loera

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MultiwayTables and Margins

0 2 2

2 1 0

4

3

2

3

2

Example:2-table of size2

X 3with line-sums:

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24

MultiwayTables and Margins

Example:3-table of size3

X 4

X 6with a plane-sum:

0

3

5

0

3

3

2

0

1

4

1

2

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MultiwayTables and Margins

0

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Example:3-table of size3

X 4

X 6with a line-sum:

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We consider two problemson the

set of all tableswith some margins fixed.

Mostly, we assume all line-sums are fixed.

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Agencies such as the census bureau and center for health statistics

allow public web-access to information on their data bases,

but areconcerned about confidentiality of individuals.

Common strategy:release marginsbut not table entries.

Question:how does theset of values that can occur in a

specific entryin all tables with the released marginslook like ?

Common perception:if the entry-rangecontainsmany values

then theentryis secure; otherwise it is vulnerable.

Problem 1 -Table Security: Entry-Range

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0 2 2

2 1

0

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2 2 0

0 1

2

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1 2 1

1 1

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2

Therefore, also the value 1occurs in that entry:

Specific example:the values 0, 2occur in an entry:

Example:for 2-tables with fixed line-sums,

theentry-rangeis always aninterval.

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Theorem 1:

Theorem 1:For everyfinite set Sof nonnegative integers,

there are r,

c

and line-sumsfor tables of size r

X

c

X

3

such that the set of values occurring in a fixed entryin all

possible tables with these line-sumsis preciselyS.

In contrast, we show the following universalitytheoerm.

Example:for 2-tables with fixed line-sums,

theentry-rangeis always aninterval.

Same holds for d-tableswith fixedhyperplane-sums.

So common practice is to compute by linear programming

lower bound

L

Land upper bound

U

Uon the possible values of

an entry and use the gap U-Las a measure of its security.

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The only valuesoccurring in that entryin all

possible tables with these line-sumsare 0,2

Example:Entry-rangewith a gap

3

2

2

1

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1

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Consider the following line-sumsfor tables of size 6

X

4

X

3:

Consider the

designated entry:

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Problem 2 -Table Sampling: Markov Bases

AMarkov basisis a set of arraysthat enables a walk between

any two tableswith thesame marginswhile staying nonnegative.

It enables samplingthe (huge) set of tableswith fixed margins.

0 0 0

0 1 -1

0 -1 1

0 0 0

1 0 -1

-1 0 1

1 0 -1

-1 0 1

0 0 0

1 -1 0

-1 1 0

0 0 0

. . .

So Markov bases of rxc

tables with fixedline-sums aresimple:

they have constant support4and constant degree1regardless of r,c.

Same holds for d-tableswith fixedhyperplane-sums.

Example:Markov bases of rxc

tables with fixedline-sums are2x2minors:

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3-tables with fixed line-sums

Markov bases of 3-tables with fixedline-sums aremuch more complicated.

In contrast, we show the following universalityof tables of size

rXcX3,

with one side3 fixedand smallest possibleandtwo sides

r,c

variable.

Theorem 2:

Theorem 2:For everyfinite set Vof integer vectors, there

are r,

c

such that any Markov basisfor r

X c

X 3tables with

fixed line-sums,restricted to some entries, containsV.

So these Markov bases have unbounded degree and support.

Nice result(Aoki-Takemura, Santos-Sturmfels):for tables of size

rXcXh

,

with two sides

c,h

fixedandone side

r

variable,there is an upper bound

u(c,h) on degreeandsupportof Markov base elements,regardless of

r.

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Toricideals and Tables

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We have the following universalitytheorem for toricideals.

The equationsforcing the same marginson tables, such as line-sums,

plane-sums, and so on, lift to a corresponding toricidealgenerated

by allbinomialscoming from pairs of tableswith the same margins:

Fundamental result (Diaconis-Sturmfels):thebinomials

xu-xv

generate a

toricidealif and only ifthe corresponding arraysu-vform a Markov basis.

Theorem 3:

Theorem 3:For everytoricideal I,there are

r,

c

such that any

generating set of the ideal of

r

X

c

X 3tables with fixed line-sums,

restricted to some variables,contains a generating set ofI.

Toricideals and Tables

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Proofs: Universality of Line-Sum Polytopes

Theorems 1-3 stated before are corollaries

of the following remarkable theorem:

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Example -Proof of Theorem 2:

Recall Theorem 2:

Recall Theorem 2:For everyfinite set Vof integer vectors,

there are r,

c

such that any Markov basisfor r

X c

X 3tables

with fixed line-sums,restricted to some entries, containsV.

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Some Problems:

Classify Markov basesof hierarchical modelsdefined by table size

n1

x

. . .

x nd

and by simplicialcomplex of supportsof fixed margins:

Implications of the universality theoremon the existenceof a

strongly polynomial timealgorithm for linear programming?

Implications on the rational version of Hilbert’s 10th

problemon

the decidabilityof the realization problem for polytopes?

-moderate, boundedmodels such as line-sumsof r

x c

x

h

tables:

-universal, intractable models such as line-sumsof r

x c

x 3tables:

-othermodels:

-simple, square-freemodels such as hyperplane-sumsof d-tables:

n1

. . .

n2

nd

r

c

h

3

c

r

n1

n

2

n

3

n

4

n

5

n

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