To prove that a polynomial is nonnegative on Rn one can try to show that it is a sum of squares of polynomials. The latter problem is now known to be reducible to a semidefinite programming computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials ...

A method for computing the multigraded Hilbert depth of a module was presented in [16]. In this paper we improve the method and we introduce an effective algorithm for performing the computations. In a particular case, the algorithm may also be easily adapted for computing the Stanley depth of the module. We further present interesting examples which were found with the help of an experimental ...

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x1, . . . , xn]. The definition involves Gröbner bases or the action of the algebraic torus (K∗)n. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter...

Multiview geometry is the study of two-dimensional images of threedimensional scenes, a foundational subject in computer vision. We determine a universal Gröbner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n − 15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fix...

The toric Hilbert scheme of a lattice L ⊆ Z is the multigraded Hilbert scheme parameterizing all ideals in k[x1, . . . , xn] with Hilbert function value one for every g in the grading monoid G = N/L. In this paper we show that if L is twodimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert...

The multigraded Poincaré-Betti series P k R(x̄; t) of a monomial ring k[x̄]/〈M〉 on a finite number of monomial generators has the form ∏ xi∈x̄ (1+xit)/bR,k(x̄; t), where bR,k(x̄; t) is a polynomial depending only on the monomial set M and the characteristic of the field k. I present a computer program designed to calculate the polynomial bR,k for a given field characteristic and a given set of monom...

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X . Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m1 1 ⊗ · · · ⊗ Bl l . We give conditions on the mi which guarantee that the ideal of X in P(H (X, L)∗) is generated by quadrics and the first p syzygies are line...

Let A be a semigroup whose only invertible element is 0. For an A-homogeneous ideal we discuss the notions of simple i-syzygies and simple minimal free resolutions of R/I. When I is a lattice ideal, the simple 0-syzygies of R/I are the binomials in I. We show that for an appropriate choice of bases every A-homogeneous minimal free resolution of R/I is simple. We introduce the gcd-complex ∆gcd(b...