Degree Bounds in Rational Sums of Squares
Representations on Curves
Greg Blekherman
Georgia Tech
FoCM 2014
Degree Bounds in Rational Sums of Squares
Representations on Curves
An exercise in proselytism
Greg Blekherman
Georgia Tech
FoCM 2014
The Question
• X ⊂ RPn projective variety with dense real points and graded
coordinate ring R[X ]. We assume that X is irreducible and
nondegenerate.
• p ∈ R[X ]2s is nonnegative on X .
The Question: What is the structure of polynomials of degree 2s
nonnegative on X ?
Reformulation: How does the geometry of X affect existence of
efficient certificates of nonnegativity?
Some Background
• Can certify nonnegativity of p by writing
p=
X qi 2
ri
, pi , qi ∈ R[X ].
• Equivalently, can find a sum of squares (of polynomials!) h
such that
p·h
is a sum of squares.
• Efficient means finding degree bounds on h.
The Theorem
PX ,2s , ΣX ,2s ⊂ R[X ]2s nonnegative polynomials and sums of
squares of degree 2s in R[X ].
Theorem:(B., G. Smith, M. Velasco) Let X ⊂ Pn be a real curve
of degree d and arithmetic genus g . Let p ∈ P2s be a nonnegative
form and let
2g − 1
k = max d − n + 1,
+1 .
d
Then there exists h ∈ ΣX ,2k such that p · h ∈ Σ2s+2k .
Remarks:
• The bound depends on simple (complex) geometric invariants
of X .
• The degree bound is independent of the degree of p.
Illustrative Example
X is planar curve of degree d. Then the degree bound on h is
2d − 4.
Let p ∈ R[X ]2s then there exists h ∈ ΣX ,2d−4 such that
p · h ∈ ΣX ,2s+2d−4 .
This bound is tight on an open subset of all planar curves of
degree d.
We don’t know of a single planar curve where this bound is not
tight!
The Journey from Past to Present
Hilbert’s 17th Problem: p ∈ R[x]2s is a polynomial nonnegative
on RPn . Is it true that p is a sum of squares of rational functions?
Artin (+Schreier) 1920’s: Yes!
Nonnegativity is semialgebraic question and it is natural to work
inside the category of semialgebraic sets.
But, no canonical presentation, e.g. can ask given
g1 , . . . , gk ∈ R[x],
let X = {x ∈ Rn | gi (x) ≥ 0, ∀i}.
Let p be a polynomial nonnegative on X , find degree bounds in
representation of p as a sum of squares of rational functions (also
using gi ).
On the Road Again
X = Pn , deg p = 2s. There exists a sum of squares multiplier h
with
n
s4
deg h ≤ 22 .
(Lombardi, Perruci, Roy, 2014).
Best known lower bound for the case deg p = 4
deg h ≥ n.
(B., Gouveia, Pfeiffer, 2014).
Good versus Evil
Affine Versions of the Question: X = {0, 1}n . Then
I (X ) = hxi2 − xi i. We have
xi = xi2 in R[X ],
so can certify nonnegativity of xi without multipliers.
Schmüdgen’s and Putinar’s Positivstellensatze.
However if dim X > 0 then the degree of the certificate depends on
the minimum of p on X .
Further Evidence
Question: What if we don’t use multipliers? When is PX2 = ΣX ,2 ?
Theorem: (B., Smith, Velasco) Let X be an irreducible,
nondegenerate real variety. Then PX ,2 = ΣX ,2 if and only if XC is
a variety of minimal degree.
Let X ⊂ CPn be a nondegenerate, irreducible variety. Then
deg X ≥ codim X + 1.
If equality is achieved then X is called a variety of minimal degree.
Classified by Del Pezzo and Bertini.
THANK YOU!