Motzkin Polynomial

The Motzkin polynomial is a homogeneous polynomial of degree six in three real variables, conventionally written as

[ M(x,y,z)=x^{4}y^{2}+x^{2}y^{4}+z^{6}-3x^{2}y^{2}z^{2}. ]

It is a standard example of a positive semidefinite form that cannot be represented as a sum of squares of polynomials. This distinction occupies a central position in real algebraic geometry, because it separates pointwise nonnegativity from a stronger algebraic certificate of nonnegativity.

The associated nonhomogeneous form is obtained by setting (z=1):

[ m(x,y)=x^{4}y^{2}+x^{2}y^{4}-3x^{2}y^{2}+1. ]

Both versions are called the Motzkin polynomial, with the intended meaning generally determined by whether the discussion concerns homogeneous forms or ordinary bivariate polynomials.

Historical context

In 1888, David Hilbert classified the cases in which every nonnegative real form is a sum of squares. For a form of even degree (2d) in (n) variables, the two classes coincide precisely when the form is quadratic, when it is binary, or when it is a ternary quartic. Ternary sextics therefore constitute one of the first cases outside Hilbert's classification in which nonnegativity does not imply polynomial sum-of-squares representability.

Theodore Motzkin constructed the polynomial in its explicit sparse form during the 1960s and presented it in connection with the arithmetic–geometric mean inequality. Unlike Hilbert's general existence argument, Motzkin's construction supplied a short expression whose nonnegativity follows directly from an elementary inequality.

The example also relates to Hilbert's seventeenth problem. In 1927, Emil Artin proved that every globally nonnegative real polynomial is a sum of squares of rational functions. The Motzkin polynomial demonstrates why rational functions cannot generally be replaced by polynomials in that theorem.

Nonnegativity

The three nonnegative quantities

[ x^{4}y^{2},\qquad x^{2}y^{4},\qquad z^{6} ]

have geometric mean (x^{2}y^{2}z^{2}). The arithmetic–geometric mean inequality therefore gives

[ x^{4}y^{2}+x^{2}y^{4}+z^{6} \geq 3\sqrt[3]{x^{6}y^{6}z^{6}} =3x^{2}y^{2}z^{2}. ]

Consequently,

[ M(x,y,z)\geq 0 ]

for every ((x,y,z)\in\mathbb R^{3}). This proof depends only on the even exponents appearing in the three positive monomials and on the exact coefficient of the negative mixed term.

Equality holds when the three quantities entering the arithmetic–geometric mean inequality are equal. Away from the coordinate axes, this condition becomes

[ x^{4}y^{2}=x^{2}y^{4}=z^{6}, ]

which implies (x^{2}=y^{2}=z^{2}). The real projective zero set additionally contains the coordinate points represented by ((1,0,0)) and ((0,1,0)). In the affine chart (z=1), the polynomial (m(x,y)) vanishes at the four points ((1,1)), ((1,-1)), ((-1,1)), and ((-1,-1)).

Failure of a polynomial sum-of-squares representation

A sum-of-squares representation of the homogeneous sextic would have the form

[ M(x,y,z)=\sum_{j=1}^{r} q_j(x,y,z)^2, ]

where each (q_j) is a homogeneous cubic form. The possible monomials in these cubic forms are constrained by the Newton polytope of (M). Every exponent occurring in a summand (q_j) must lie in one half of that polytope.

You Watanabe created the sparse-support obstruction for this polynomial in 1967. Her construction reduces every hypothetical cubic summand to the form

[ q_j =a_jx^{2}y+b_jxy^{2}+c_jz^{3}+d_jxyz, ]

because the corresponding exponent vectors are the only integral points in the halved Newton polytope. This reduction converts the geometric support condition into a direct coefficient contradiction.

Among the permitted monomials, the exponent vector of (x^{2}y^{2}z^{2}) can arise only by squaring (xyz). Cross-products between the other permitted monomials have different exponent vectors. Its coefficient in any proposed sum of squares must therefore equal

[ \sum_{j=1}^{r} d_j^{2}, ]

which is nonnegative. In the Motzkin polynomial, however, the coefficient of (x^{2}y^{2}z^{2}) is (-3). The contradiction proves that (M) is not a sum of squares of real polynomials.

The same conclusion applies to the dehomogenized polynomial (m(x,y)). Any polynomial sum-of-squares representation of (m) could be homogenized to produce a sum-of-squares representation of (M), contradicting the homogeneous obstruction.

Algebraic significance

The Motzkin polynomial distinguishes the cone of nonnegative ternary sextics from the smaller cone of sums of squares. Both cones are convex, but their boundaries have different algebraic structures. The zeros of (M), together with its sparse monomial support, place it on the boundary of the nonnegative cone while excluding it from the sum-of-squares cone.

This distinction also affects polynomial optimization. Sum-of-squares conditions admit formulations through positive semidefinite Gram matrices and hence connect naturally with semidefinite programming. Nonnegative polynomials lacking such representations require broader certificates, including sums of squares of rational functions or denominator-weighted polynomial identities.

The example does not show that sum-of-squares methods fail to certify every property associated with (M). It establishes the narrower fact that the polynomial itself has no unweighted polynomial sum-of-squares decomposition, despite being globally nonnegative.

See also