POSITIVE POLYNOMIALSweb.math.ucsb.edu/~mputinar/POSAMS.pdf · 4 1. Minkowski and Hilbert Every positive integer is the sum of four squares LAGRANGE Academie des Sciences de Paris,

POSITIVE POLYNOMIALS

MIHAI PUTINAR

AMS at UCSB April 2005.1

2

The problem of the representation of an inte-ger n as the sum of a given number k of integralsquares is one of the most celebrated in the his-tory of numbers. Its history may be traced backto Diophantus, but begins effectively with Girard’s(or Fermat’s) theorem that a prime 4m + 1 is thesum of two squares. Almost every artihmeticianof note since Fermat has contributed to the solu-tion of the problem, and it has its puzzles for usstill.

G.H.HARDY, 1940

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.

Contents

1. Minkowski and Hilbert 42. Functional analysis 103. Real algebra and logic 144. The Positivstellensatz 185. Modern real algebraic geometry 236. Moment problems 257. Optimization theory 298. Real free *-algebra 329. Complex variables 35

4

1. Minkowski and Hilbert

Every positive integer is the sum of four squaresLAGRANGE

Academie des Sciences de Paris, 1881: ”Study the(number of) decompositions of an integer into fivesquares”

MINKOWSKI, 1882: Memoire sur la theorie desformes quadratiques a coefficients entiers

wins the Grand Prix (the author is 17 years old!)

” The decomposition of an integer into five squaresdepends on quadratic forms of four variables, just asthe representation of an integer as a sum of threesquares depends on quadratic forms in two variables,as shown by Gauss.”

5

Inaugural Dissertation:MINKOWSKI, (Konigsberg, 1885) with Hilbert

as opponent, published as:

Untersuchungen uber quadratische formen, ActaMath. 7(1886), 201-256.

Minkowski’s thesis: ”It is not probable thatevery positive form can be represented asa sum of squares”

clarified and supported by Hilbert

Uber die Darstellung definiter formen als Sum-men von Formenquadraten, Math. Ann. 32(1888),342-350

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dim. 1

p(x) ≥ 0 for all x ∈ R

p(x) = q(x)2∏k

[rk(x)2 + sk(x)2]

and

(a2 + b2)(c2 + d2) = (ac− bd)2 + (ad + bc)2

or better

|a + ib|2|c + id|2 = |(a + ib)(c + id)|2,imply

p(x) = p1(x)2 + p2(x)2.

7

dim. 2

Hilbert’s construction:

Choose 9 points a1, . . . a9 ∈ R2 on two cubics, sothat for every polynomial Q:

degQ = 3, & Q(ak) = 0(1 ≤ k ≤ 8) ⇒Q(a9) = 0.

Let P (x, y) ≥ 0 on (x, y) ∈ R2,

degP = 6 and

P (ak) = 0, 1 ≤ k ≤ 8, but P (a9) > 0.

Then it is impossible to have

P =∑

j

Q2j .

8

1900, HILBERT’s 17-th problem:

”Expression of definite forms by squares:

...the question arises whether every definite formmay not be expressed as a quotient of sums ofsquares of forms...... it is desirable, for certain questions as to the

possibility of certain geometrical constructions, toknow whether the coefficients of the forms to beused in the expression may always be taken fromthe realm of rationality given by the coefficientsof the form represented.”

9

1909, HILBERT solves Waring’s problem using acubature formula with rational coefficients for

(x21+...+x2

5)m = C

∫|t|≤1

(t1x1+...+t5x5)2mdt1...dt5.

1912 HILBERT publishes the book ”Grundzugeeiner allgemeinen Theorie der Linearen Integral-gleichungen”

based on six independent, and previously publishedarticles (1904-1910).

Background theme: spectral analysis requires in-finite sums of squares (SOS) decompositions, ofpossibly infinitely many variables.

Based on concrete equations of mathematical physicsand problems of function theory.

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2. Functional analysis

Hilbert-Hahn-Hellinger- spectral theorem:

A = A∗ linear bounded Hermitian operatorp ∈ R[x], p|R ≥ 0 ⇒ p(A) ≥ 0.

Proof. p =∑

q2k hence

p(A) =∑

qk(A)∗qk(A) ≥ 0.

Conclusion:

p 7→ p(A)

is a monotonic, bounded (1 7→ I) calculus. It can beextended to all bounded measurable functions, andrepresented by a positive, operator valued, measure:

p(A) =

∫R

xEA(dx)

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Fejer-Riesz factorization:

p(eiθ) ≥ 0, (θ ∈ R) ⇒ p(eiθ) =∑

k

|qk(eiθ)|2

and one can choose q(eiθ) =∑N

j=0 cjeijθ.

Application:

U−1 = U ∗, p(eiθ) ≥ 0 ⇒ p(U) ≥ 0.

Has many surprising applications, for instance(Fejer’s inequality):

p(eiθ) =

d∑j=−d

pjeijθ ≥ 0 ⇒

|p1| ≤ p0 cosπ

d + 2.

12

Riesz-Herglotz formula

p ∈ C[z], <p(z) ≥ 0 (|z| ≤ 1)

can be written:

p(z) = i=p(0) +1

2π

∫ π

−π

eiθ + z

eiθ − z<p(eiθ)dθ.

Consequently, denoting dµ(eiθ) = <p(eiθ)dθ2π :

p(z) + p(w)

1− zw=

∫|u|=1

dµ(u)

(1− uz)(1− uw)

= 〈 1

1− uw,

1

1− uz〉2,µ

≈∑

k

ck

(1− ukw)(1− ukz),

with ck > 0.

Non-negative definite kernel as a continuous SOS,or discrete, but infinite SOS.

Thus, any non-negative harmonic polynomial inthe unit disk D can be decomposed as:

<p(z) = (1− |z|2)∞∑

k=1

|fk(z)|2,

with fk ∈ O(D).

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Stieltjes, Schur, Nevanlinnaanalyticity + SOS (= positivity assumption) has an

additional and very useful continued fraction struc-ture. Example:

If p : D −→ D, then

p(z) =zr1(z) + γ0

1 + γ0zr1(z)

and degr1 < degp,...

Clarifies classical bounded interpolation problemsfor analytic functions.

Kolmogorov, 1940Let X be a set and K : X ×X −→ C a positive

semi-definite kernel, that is

(K(xi, xj))i,j∈I ≥ 0,

as a matrix, for all finite subsets I ⊂ X .Then there exists a Hilbert space H and a map

k : X −→ H such that

K(x, y) = 〈k(x), k(y)〉, x, y ∈ X.

Therefore

K(x, y) = 〈∑

j

kj(x)ej,∑m

km(y)em〉

=∑

j

kj(x)kj(y).

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3. Real algebra and logic

E. Artin, Uber die Zerlegung definiter Funktio-nen in Quadrate, Abh. math. Sem. Hamburg5(1926), 100-115.

E. Artin, O. Schreier, Algebraische Konstruk-tion reeler Korper, Abh. math. Sem. Hamburg5(1926), 85-99.

Extensions of real fields, and of positive cones (or-ders).

The following are equivalent:

a). K is a real closed field;b). K2 is a positive cone and every odd degree

polynomial in K[X ] has a root in K;c). K 6= K(

√−1) and K(

√−1) is algebraically

closed.

Every ordered field K has a real closure, unique upto a K-isomorphism.

Ideas originating in Hilbert’s Foundations of Ge-ometry.

Application: Every non-negative polynomial is aSOS of rational functions.

SOS in K are totally positive elements, i.e. posi-tive w.r. to every ordering of K.

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Sturm’s theorem

Start with (f, f ′) = 1 and define fi, gi ∈ R[X ] as:

f0 = f, f1 = f ′,

f0 = g1f1 − f2, degf2 < degf1,...

fm−2 = gm−1fm−1 − fm, degfm = 0.

Consider an interval [a, b] ⊂ R, such that fj(a)fj(b) 6=0 for all j.

Let N(x) be the number of sign changes in thesequence f0(x), f1(x), . . . , fm(x). Then

The number of roots of f in [a, b] is N(a)−N(b).

The same result holds in every real closed field.

Refined by Sylvester.

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Tarski’s elimination theory for real closed fields

To any formula φ(X1, ..., Xn) in the vocabulary0, 1, +, ·, < and with variables in a real closedfield, one can effectively associate two objects:(i) a quantifier free formula φ(X1, ..., Xn) in the

same vocabulary, and(ii) a proof of the equivalence φ ≡ φ that uses

only the axioms of real closed fields.

Completeness of elementary algebra and geometry,based on an effective decision method. If valid inone real closed field, then it is valid in all other. Idealater developed by Lefschetz in his transfer principle.

Main question: does the system

f (X) = 0, g1(X) > 0, ..., gk(X) > 0

admit a solution in a real closed field R, based on acriterion which is rational in the coefficients of thepolynomials f, g1, ..., gk.

Lectures in 1927, announced in Ann. Soc. Pol.Math. 9(1931), published in Fund. Math. 17(1931);full manuscript of 1940 published by 1948 (RAND);Inst. B. Pascal 1967.

Simplification by Seidenberg: A new decision methodfor elementary algebra, Ann. Math. 60(1954), 365-374.

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Applications came late, but were spectacular.

Hormander’s inequality (1955):

For each polynomial f (X1, ..., Xn) ∈ R[X1, ..., Xn]there are positive constants c, r such that

|f (x)| ≥ c dist(x, V (f ))r, x ∈ Rn, |x| ≤ 1.

Generalized to real analytic functions by Lojasiewicz(1964).

Division of distributions by real analytic functions,i.e. elementary solutions to constant coefficient linearPDE’s. Hypoellipticity...

————————-

In practice:

A basic semi-alegbraic set

x ∈ Rn; f (x) = 0, q1(x) > 0, ..., qk(x) > 0has semi-algebraic projections.

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4. The Positivstellensatz

Simplest decision problem:

x2 + 2bx + c > 0, x ∈ R,

is equivalent to:b2 < c.

Then one can complete the squares

x2 + 2bx + c = (x + b)2 + (c− b2).

See other similar root separation criteria (Routh,Hurwitz, Cohen, Schur, Takagi).

———————–

Hilbert’s Nullstellensatz:

x ∈ Cn; f1(x) = 0, ..., fk(x) = 0 = φ

with fj in the polynomial ring C[x] is equivalent to

g1f1 + ... + gkfk = 1

for some gj ∈ C[x].

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Krivine Anneaux preordonnes, J. Analyse Math.12(1964), 307-326.

Dubois A nullstellensatz for ordered fields, Ark.Mat. 8(1969), 111-114.

Stengle A Nullstellensatz and a Positivstellen-satz in semialgebraic geometry, Math. Ann. 207(1974),87-97.

A subset A ⊂ R[x] generates:I(A) - an ideal (

∑aipi)∏

(A) - a multiplicative monoid (with 1) (aiaj...am)T (A) - a convex subsemiring (

∑ai...akp

2).

Let F, G, H ⊂ R[x]. The set of Rn given by f (x) = 0 (f ∈ F ),g(x) 6= 0 (g ∈ G),h(x) ≥ 0 (h ∈ H),

is empty if and only if there are elementsa ∈ I(F ), b ∈

∏(G), c ∈ T (H) such that

b2 + c = a.

R[x] can be replaced by any commutative ring with1 and evaluations at x ∈ Rn by evaluations at pointsof the real spectrum.

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Examples:

Assume p ∈ R[x] is non-negative on Rn. Then

x ∈ Rn; p(x) 6= 0,−p(x) ≥ 0 = φ.

Thus there exists b ∈∏

(p), c ∈ T (−p) such that:

b2 + c = 0,

that is:p2k + f 2 − pg2 = 0,

p =p2k

g2+

f 2

g2.

——————

f = 0 on g = 0 implies:

f 2k + SOS ∈ (g).

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Back to the methods of Minkowski and Hilbert:

G. Cassier (1984), K. Schmudgen (1991),M.P. (1991-93).

Let S = x ∈ Rn; Pi(x) ≥ 0, 1 ≤ i ≤ N, with P1

strictly negative at infinity, be compact and assumef > 0 on S. Then

f ∈ SOS + P1 SOS + ... + PN SOS.

No denominators, and no terms of the form Pi...Pk SOS.

Algebraic proof: T. Jacobi, A. Prestel (2001).

Example.

f > 0 on the first quadrant of the unit ball implies

f (x) ∈ SOS+(1−‖x‖2)SOS+x1SOS+...+xnSOS.

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Sketch of proof:Assume the contrary and separate (a la Minkowski)

f from C = SOS + P1 SOS + ... + PN SOS by alinear functional:

L : R[x] −→ R, L(f ) < 0 ≤ L|C.

—————–

L(hg) = 〈h, g〉Lis a positive semi-definite inner product; can be com-pleted to a Hilbert space, and

〈xjh, g〉L = 〈h, xjg〉L

x = (x1, ..., xn) strongly commuting system of sym-metric operators, subject to Pj(x) ≥ 0. Hence x hascompact spectrum, hence it is bounded.

Spectral representation:

〈F (x)1,1〉 =

∫S

Fdµ

implies

0 > L(f ) = L(f1) = 〈f (x)1,1〉L =

∫S

fdµ ≥ 0,

a contradiction.

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5. Modern real algebraic geometry

The real spectrum (M. Coste, 1979)

Let A be a commutative ring with 1

Sper(A) = P; P positive cone .

P ⊂ A is a positive cone if:

P + P ⊂ P, P.P ⊂ P, A2 ⊂ P, −1 /∈ P,

and P ∪ −P = A, P ∩ (−P ) is a prime ideal.

Example: Fix a ∈ R and definePa = p ∈ R[X ]; p(a) ≥ 0.

Many modern ramifications:

- dimension theory, finite maps- intersection theory, enumerative aspects- stratification and triangulation- topology of real algebraic varieties (Hilbert’s 16-th

problem)- vector bundles, Pic, K-theory, Witt ring- constructive aspects and algorithmic/complexity

problems- subanalytic sets, constructible sheaves

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Examples(Scheiderer, 2004)

Let Y be a smooth affine curve over R (real closedfield) which is rational. Then every element f ∈R[Y ] which is non-negative is a sum of squares.

Let Y be a smooth connected affine curve of genusg ≥ 1 which has only real points at infinity. Thenthere exists f ∈ R[Y ], f ≥ 0, which is not a sum ofsquares.

——————————Hilbert (1888)Every real homogeneous polynomial p(x, y, z) of

degree 4 is a sum of three squares of quadraticforms.

Powers, Reznick, Scheiderer, Sottile (2004)If in addition the plane curve defined by p is

smooth, then there are exactly 8 non-equivalentsuch representations.

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6. Moment problems

Dual interpretation of SOS

Let A = R[X ] be the algebra of regular functionson a real variety X . A moment functional

L : A −→ R,

satisfies L(f 2) ≥ 0, f ∈ A.

Link to spectral theorem: the Hermitian product

〈f, g〉 = L(fg)

produces a Hilbert space H and each f ∈ A acts asan unbounded multiplier

Mf : D(f ) ⊂ H −→ H.

Basic questions, going back to mid XIX-th century:

1. Is there a positive measure µ on X such that:

L(f ) =

∫fdµ, f ∈ A?

2. If so, is this measure unique?

Quintessential inverse problem, still of interest forits many applications to probability theory, signalprocessing, tomography, geophysics, image reconstruc-tion, ...

26

M. Riesz (1921): A positive extension

L : C(X) −→ R, L|C+(X) ≥ 0, LA = L,

would solve 1. Problem 2 is a density, or complete-ness question.

Constructive aspects:One starts with X ⊂ Rn and with the data (mo-

ments):aα = L(xα), α ∈ Nn.

The orthogonal polynomials

pα(x) = xα +∑β<α

cβxβ

are recursively computable, and Mxjbecomes a Ja-

cobi matrix . . . .

Was refined by von Neumann into his theoryof unbounded symmetric operators and their self-adjoint extensions.

27

Examples:

an = L(xn), n ≥ 0,

is a moment functional if and only if the Hankel ma-trix (am+n)∞m,n=0 is positive semi-definite. Then L al-ways admits a representing positive measure, (Ham-burger, 1919).

Uniqueness criterium (Carleman, 1922):∑n

1

a1/2n2n

= ∞.

——————————

X = z ∈ C; |z| = 1, and

an = L(zn), n ≥ 0.

If L is real then a−n = an.

0 ≤ L(|∑

snzn|2) =

∑m,n

L(zn−m)snsm

implies ∑m,n

an−msnsm ≥ 0.

Direct harmonic analysis connection:

an = L(zn) =

∫ π

−π

einθdµ(θ).

Studied by Toeplitz and Caratheodory (1911),...

28

Back to Hilbert’s problem: universal denominators.

Assume that p(x) ≥ 0, x ∈ Rn. Then there existsN ∈ N and polynomials gk such that

(1 + |x|2)Np(x) =∑

k

gk(x)2.

and similarly for weighted SOS decompositions (withprescribed supports).

Reznick (1996) algebraic-combinatorial proof

MP-Vasilescu (1999) Hilbert space approach, basedon von Neumann’s theory of extensions of symmetricoperators

Scheiderer, 2005: Complexity of the moment prob-lem

In general, in the decomposition

p ∈ SOS +∑

qkSOS

the degrees of the summands cannot be bounded byfunctions depending on the degree of p.

29

7. Optimization theory

Let S ⊂ Rn be a (basic) semi-algrebaic set and letp ∈ R[x].

Claim:p∗ = minp(x); x ∈ S

is equivalent to

p(x)− p∗ ∈ SOS (mod S).

Too nice to be true. How false it can be?

If S is compact, then always true, for p(x)− p∗ +ε, ε > 0 (Schmudgen, MP, 1991-1993). In general,not true.

Great numerical advantage, because, when true,SOS decompositions can be checked with semidefi-nite programming, that is optimization over the coneof all positive definite matrices.

N.Z.Shor, Class of global minimum bounds forpolynomial functions, Cybernetics 23(1987), 731-734.

Y.E.Nesterov, A. Nemirovski, Interior point meth-ods in convex programming, SIAM, 1994.

P. Parrilo, Structured semidefinite programs andsemialgebraic geometry methods in robustness andoptimization, PhD Thesis, 2000.

30

P.Parrilo, B. Sturmfels Minimizing polynomialsfunctions, DIMACS Series in Discrete Math., 2001.

B. Sturmfels, J. Demmel and J. Nie, Minimizingpolynomials via sum of squares over the gradientideal, U.C. Berkeley, 2005.

However,

G. Blekherman, There are significantly more non-negative polynomials than sums of squares, U. Michi-gan, 2004.

(obtained by computing volumes of the respectivecones, for a fixed degree)

In spite of the theoretical limitations, the applica-tions abound and are on the rise.

More general, related problems:

S. Basu, R. Pollack, M.-F. Roy, Algorithms in realalgebraic geometry. Berlin Heidelberg New York:Springer 2003.

31

Jean Lasserre’s relaxation method:

Global optimization with polynomials and theproblem of moments, SIAM J. Optimization 11(2001),796-817.A new hierarchy of SDP-relaxations for polyno-

mial programming, Toulouse, 2005.

Instead of minimizing

p(x), x ∈ S,

consider

min

∫S

p(x)dµ(x),

over all probability measures µ on S.Use the moments

yα =

∫S

xαdµ(x), α ∈ Nn,

as auxiliary variables.

Always convergent, and it allows an optimal as-ymptotic/error analysis

Software:

GloptiPoly www.laas.fr/∼lasserreSOSTOOLS www.cds.caltech.edu

32

8. Real free *-algebra

Let F = R〈x1, ..., xn, x∗1, ..., x

∗n〉 be the free alge-

bra with R-linear involution

(xi)∗ = x∗i , (x∗i )

∗ = xi,

(fg)∗ = g∗f ∗, f, g ∈ F.

Represented on n-tuples of matrices X = (X1, ..., Xn)acting on a (finite dimensional) real Hilbert space H :f 7→ f (X).

Let

S = (X1, ..., Xn) ∈ L(H)n; X∗1X1+...+X∗

nXn = 1be the non-commutative sphere, with associated ideal

I(S) = f ∈ F; f (X) = 0, X ∈ S.

f (X) ≥ 0 means positive semi-definite as a matrix.

33

J.W. Helton, Positive non commutative polyno-mials are sums of squares, Ann. Math. 56(2002),675-694:

f ∈ F, f (X) ≥ 0 for all X ∈ L(H)n, then thereare finitely many gk ∈ F:

f =∑

g∗kgk.

Helton, McCullough, MP, A non-commutative Pos-itivstellensatz on isometries, J. reine angew. Math.568(2004), 71-80:

f ∈ F, f(X) ≥ 0 for all X ∈ S, then there arefinitely many gk ∈ F:

f =∑

g∗kgk (mod I(S)).

Also motivated by real world problems (dimension-less linear control theory questions). Proofs basedon the same Minkowski separation argument, plusCaratheodory’s theorem (about convex hulls in Rn).

www.math.ucsd.edu/∼heltonsoftware NC-ALGEBRA

34

Recent variations and adaptations to the Weyl al-gebra, Jordan algebras, enveloping algebras of solv-able Lie algebras.

F. Radulescu, 2005: Connes embedding conjecture(every type II1 factor can be embedded into the fac-tor Rω) is equivalent to a SOS-type decompositionof polynomials p ∈ F satisfying

tr p(X) ≥ 0, X = X∗.

35

9. Complex variables

Under some generic conditions, a domain in Cn

given by the polynomial equation:

Ω = z ∈ Cn;

n∑k=1

|Pk(z)|2 < 1

covers the unit ball B, via the finite fibre map:

(P1, P2, ..., Pn) : Ω −→ B.

Global, or local, converses of this statement, pos-sibly twisted by a biholomorphic automorphism, arecrucial in modern complex geometry.

D. Quillen, On the representation of Hermitianforms as sums of squares, Invent. Math. 5 (1968),237-242:

If a bi-homogeneous polynomial p ∈ C[z, w] satis-fies p(z, z) > 0 for z 6= 0, then there exists N ∈ Nsuch that:

|z|2NP (z, z) =∑

k

|Pk(z)|2.

Refined and geometrized by D. Catlin and J. D’Angelo(1997-2005).

Similar representations appear naturally in pluri-complex potential theory, Fantappie transforms, com-plex Radon transforms, ...

36

ExampleMP Sur la complexification du probleme des mo-

ments, C. R. Acad. Sci. Paris, t. 314, Serie I(1992),743-745.

Let p(z, z) be a strictly positive (pluriharmonic)polynomial on B (or a semialgebraic support). Then

p(z, z) =∑

k

|Qk(z, z)|2 + (1− |z|2)∑

j

|Pj(z)|2.

However, if n ≥ 2, then

p(z, z) 6=∑

k

|Qk(z)|2 + (1− |z|2)∑

j

|Pj(z)|2,

even if p(z, z) ≥ 0.

Explanation: Again Hilbert space (subnormal com-mutative tuples of operators) and

Mult H(1

1− z.w) 6= H∞(B).

37

A. Tarski: A decision method for elementary al-gebra and geometry, Univ. California Press, Berke-ley, 1951.

”By a decision method for a class K of sentences(or other expressions) is meant a method by means ofwhich, given any sentence Θ, one can always decidein a finite number of steps whether Θ is in K...

The importance of the decision problem for thewhole of mathematics (and for various special math-ematical theories) was stressed by Hilbert, who con-sidered this as the main task of a new field of math-ematical research for which he suggested the term”metamathematics”. ”