A study of nonlinear multiview varieties

We study the nonlinear generalization of the classical multiview variety, which is a fundamental concept in computer vision. In this paper, we take the first comprehensive step to develop the nonlinear analogue of multiview varieties. To this end, we introduce a multigraded version of the saturated special fiber ring. By applying this tool, we are able to compute the multidegrees of several families of nonlinear multiview varieties.


INTRODUCTION
Motivation.Let k be a field.The classical multiview variety is an important tool in computer vision, see [2].It is defined as the closure of the image Y of a linear rational map G : Conceptually, the image of G is a sequence of cameras that take 2-dimensional images of 3-dimensional space.Naturally, properties about Y are crucial to solving the problem of reconstructing a 3-dimensional object from a collection of images.
Since G is linear, the defining ideal J of Y has a structure that is amenable to study via a number of different methods.Firstly, J is an example of a Cartwright-Sturmfels ideal, see [2,21,22].And so, J satisfies a number of pleasant properties, see [21].Secondly, it is possible to give an explicit description of the generators of J, see [1].In particular, if the focal points of the cameras are distinct then J is generated by certain determinants of degree 2 and 3, see [1,Theorem 3.7].Thirdly, one can study the structure of the coordinate ring of Y in terms of products of linear ideals, see [23].In particular, it is possible to give an explicit formula for the Betti numbers in terms of the polymatroidal data defined by the positions of the cameras.These results demonstrate connections between properties of Y and properties of the base locus of G.The goal and purpose of this paper is to extend such connections for whose image can be thought of as the nonlinear generalization of a multiview variety.Note that, the closure of the image Y of G naturally admits a Z p -grading, and so we will be concerned with the multidegrees of Y instead of the degree.Our first main result is Theorem 3.6 which relates the mixed multiplicities, see Definition 2.3, of the saturated special fiber ring with the multidegrees of Y.In particular, if the special fiber ring is integrally closed, we obtain a criterion for birationality of G in terms of the saturated special fiber ring.
Our second main result is Theorem 3.8 which gives an upper bound for the multidegrees of Y, and, if the base locus of G is zero-dimensional, then it yields an exact formula for the multidegrees of Y in terms of the mixed multiplicities of the base points.This generalizes the well-known degree formula for rational maps with finite base locus to the multiprojective setting, see [9,Theorem 2.5], [54,Theorem 6.6] and [8,Theorem 3.3].Note that, if G is a linear rational map, then Y is multiplicity-free, see Theorem 4.2 which recovers [47,Theorem 1.1] and [23,Theorem 3.9].
Our third main result is Theorem 5.5 which gives an explicit formula for the multidegrees of Y in the case that the base ideals I 1 , . . ., I p−1 are zero-dimensional and I p is either a height two perfect ideal or a height three Gorenstein ideal.In particular, this result generalizes [14, Theorem A] and [19,Theorem A] to the multigraded setting.
Lastly, for the case of monomial rational maps, Theorem 6.4 provides a closed formula for the multidegrees of Y in terms of mixed volumes of some naturally constructed lattice polytopes.
Outline.In Section 2, we fix our notation and recall the definitions of multidegrees and mixed multiplicities, see Definition 2.3.In Section 3, we begin by fixing Setup 3.1 which details the generality for which our main results hold and, following this, we recall some fundamental properties of the multigraded Rees algebra.In §3.2, we introduce our main tool for studying the multidegrees of the image of rational maps, namely the saturated special fiber ring, see Definition 3.4.Using this tool, we state and prove our first main result: Theorem 3.6.In §3.4,we define the mixed multiplicities of the base locus of G and prove our second main result: Theorem 3.8, which allows us to compute the multidegrees of Y in terms of the mixed multiplicities of the base locus of G.
We proceed to apply Theorem 3.8 to important families of examples.In Section 4, we show that our results recover previous results for linear rational maps, i.e. for classical multiview varieties.In Section 5, we prove our third main result, Theorem 5.5, which computes the multidegrees of Y for a large family of rational maps extending previous work in [14,19] to the multigraded setting.In Section 6, we consider monomial rational maps G, and we show that in this case, the multidegrees of the image can be computed in terms of the mixed volumes of certain lattice polytopes.
In Section 7, we further elaborate on the previous works, along with interpretations and applications related to our results.

2.1.
Multigraded schemes and multidegrees.We fix the following setup during this subsection.
Setup 2.1.Let k be a field and S a finitely generated standard N p -graded algebra, i.e. [S] 0 = k and S is finitely generated over k by elements of degree e i with 1 ⩽ i ⩽ p.
Given a standard N p -graded k-algebra S, we consider the corresponding multigraded scheme.
We embed X = MultiProj(S), as a closed subscheme, into the multiprojective space P : Under the notation of (1), we define the following invariants.Whenever the chosen embedding of X into a product of projective spaces is clear from the context, we simply write deg d (X) instead of deg d P (X).In classical geometrical terms, when k is algebraically closed, deg d P (X) is also equal to the number of points (counting multiplicity) in the intersection of X with the product [57] and [16,Theorem 4.7].

Mixed multiplicities of ideals.
Let us recall the definition of mixed multiplicities of ideals in the setting that we are considering.For more details, we refer the reader to [55].Let R = k[x 0 , . . ., x r ] be a standard graded polynomial ring, m = (x 0 , . . ., x r ) the graded irrelevant ideal, and I 1 , . . ., I p ⊂ R a sequence of homogeneous ideals.We consider the N p+1 -graded k-algebra The mixed multiplicities of the ideals I 1 , . . ., I p with respect to m are defined as the mixed multiplicities of the algebra T (m | I 1 , . . ., I p ).

OUR MAIN TOOL: THE SATURATED SPECIAL FIBER RING
In this section, we develop and study the multigraded saturated special fiber ring.Originally, the saturated special fiber ring was introduced in [8], and has been successfully used to study several classes of rational maps [10, 14-16, 19, 20].
We begin by fixing the following setup, which is used throughout this section.
Setup 3.1.Let k be an arbitrary field, R = k[x 0 , . . ., x r ] a standard graded polynomial ring and m = (x 0 , . . ., x r ) ⊂ R the graded irrelevant ideal of R. For each 1 ⩽ i ⩽ p, let {f i,0 , f i,1 , . . ., f i,m i } ⊂ R be a set of forms of the same degree δ i ⩾ 1, I i = (f i,0 , f i,1 , . . ., f i,m i ) ⊂ R the ideal generated by these forms and G i : P r k P m i k the rational map determined by (x) = (x 0 , . . ., x r ) → (f i,0 (x), f i,1 (x), . . ., be the rational map given by the product of G 1 , . . ., the closure of the graph of G. Let P be the multiprojective space With respect to the above setting, we call the nonlinear multiview variety associated to G. Note that, if δ i = 1 for all i, then we are in the classical case of multiview varieties.

3.1.
Rational maps and some related algebras.In this subsection, we recall some basic facts regarding rational maps.The (multigraded) Rees algebra of the ideals I 1 , . . ., I p is given by Let A be the standard N p+1 -graded polynomial ring where deg(x i ) = e 1 ∈ N p+1 for all 0 ⩽ i ⩽ r, and deg(y i,j ) = e i+1 ∈ N p+1 for all 1 ⩽ i ⩽ p, 0 ⩽ j ⩽ m i .By setting deg(t i ) = −δ i e 1 + e i+1 ∈ N p+1 for all 1 ⩽ i ⩽ p, the Rees algebra R(I 1 , . . ., I p ) inherits a natural N p+1 -graded structure.And so, we can present R(I 1 , . . ., I p ) as quotient of A via the following N p+1 -homogeneous surjective R-homomorphism Thus, the N p+1 -graded structure of R(I 1 , . . ., I p ) is given by where each N p+1 -graded part is Since we are primarily interested in the R-grading of the Rees algebra, we fix the following notation.The (multigraded) special fiber ring of the ideals I 1 , . . ., I p is given by Note that F(I 1 , . . ., I p ) is naturally a standard N p -graded k-algebra.As the ideals I i 's are equigenerated in degrees δ i 's, Nakayama's lemma yields the natural isomorphism F(I 1 , . . ., I p ) ∼ = R(I 1 , . . ., I p )⊗ R R/m.In the following proposition, we recall the known relationships between the rational map G and the algebras R(I 1 , . . ., I p ) and F(I 1 , . . ., I p ). (ii) The closure of the graph of G coincides with Γ = MultiProj(R(I 1 , . . ., I p )).
Proof.For details, see e.g.[20, §3].□ We will now recall some equivalent conditions for a rational map to be generically finite.For any homogeneous ideal J ⊂ R, we denote by ℓ(J) := dim (R(J)/mR(J)) the analytic spread of J.We have the following equality dim The Segre embedding yields the isomorphism Thus, we obtain that dim(Y) = ℓ(I 1 • • • I p ) − 1.Finally, we say that the rational map G is generically finite when any of the following equivalent conditions is satisfied: Whenever G is generically finite, we define the degree of G : The map G is said to be birational if deg(G) = 1.If the degree of the field extension K(P r k )|K(Y) is infinite, then we say that G has no well-defined degree and, by convention, we write deg(G) = 0.

3.2.
A multigraded version of the saturated special fiber ring.Our main tool to study the rational map is the following algebra.
Definition 3.4.The (multigraded) saturated special fiber ring of the ideals I 1 , . . ., I p is the N p -graded k-algebra given by An important result regarding the saturated special fiber ring is the following.
Proof.The proof is analogous to the proof of [8,Proposition 2.7].□ Our main result is the following theorem, which says that the mixed multiplicities of the saturated special fiber ring are equal to the product of the multidegrees of Y and the degree of G.This provides an extension [8, Theorem 2.4] for nonlinear multiview varieties.Theorem 3.6.Assume Setup 3.1 and suppose that G : is generically finite.Then, the following statements hold: (ii) Under the additional condition of F(I 1 , . . ., I p ) being integrally closed, G is birational if and only if F(I 1 , . . ., I p ) = F(I 1 , . . ., I p ).
Proof.(i) For ease of notation, let S := F(I 1 , . . ., I p ) and T := F(I 1 , . . ., I p ).We consider the following functions These functions coincide with the top-degree parts of the Hilbert polynomials of S and T .In other words, we can write Since we have the equalities and it follows that the functions F S and F T can be obtained by computing the multiplicities of the special fiber ring and the saturated special fiber ring of products of powers of the ideals I i 's.More precisely, we have that + , we compose Veronese and Segre embeddings to obtain the following natural embedding Hence, we obtain the following commutative diagram is the rational map induced by the set of generators naturally obtained for the ideal [8,Theorem 2.4] may be applied to the rational map G ′ n and gives the following equalities Finally, by comparing the coefficients of the polynomials F S (n) and F T (n), the claimed equality follows for all d ∈ N p with |d| = r.

3.3.
A cohomological formula relating the degree of the rational map and the multidegrees of the image.Throughout this subsection, we continue using Setup 3.1.Here, we give a formula that relates the degree of G with the mixed multiplicities of H 1 m (R(I 1 , . . ., I p )) 0 and the multidegrees of the image Y. Notice that H 1 m (R(I 1 , . . ., I p )) 0 is a finitely generated N p -graded module over the special fiber ring F(I 1 , . . ., I p ), and so we can consider its mixed multiplicities.The following result extends [8, Corollary 2.12] into the current setting.
Corollary 3.7.Assume Setup 3.1 and suppose that G : is generically finite.Then, for all d ∈ N p with |d| = r, we have the equality In particular, G is birational if and only if dim Supp H 1 m (R(I 1 , . . ., I p )) 0 ∩ Y < r.
Proof.We have the short exact sequence 0 → which can then be written as The additivity of mixed multiplicities gives the equality e(d, F(I 1 , . . ., and so the result follows from Theorem 3.6.□

3.4.
A degree formula for rational maps.Here, we study the case when is not defined only in a finite number of points.In particular, we provide a formula that relates the degree of G, the multidegrees of the image, and the mixed multiplicities of the base points.This can be seen as an extension of a known degree formula, see e.g.[9, Theorem 2.5], [54,Theorem 6.6] and [8,Theorem 3.3]).This formula could also be derived by utilizing [33,Proposition 4.4].We also give a general upper bound for the degree of the image for arbitrary rational maps that are generically finite.
Hereafter, we use the same notations and conventions of Setup 3.1.We have that the base locus of G (i.e. the points where G is not well-defined) is given by We then have that dim(B(G)) = 0 (i.e. the base locus is finite) if and only if dim(R/I i ) ⩽ 1 for all 1 ⩽ i ⩽ p.Assuming that dim(B(G)) = 0, we then have the equalities for all n 1 , . . ., n p ∈ N, where κ(p) denotes the residue field of the local ring O P r k ,p .Since we have dim(R p /I i R p ) = 0 by assumption, it follows that the function For further details, we refer the reader to the survey [55] and Chapter 17 of the book [38].The polynomial P R p ;I 1 ,...,I p (n 1 , . . ., n p ) is of degree r if there is some i with I i R p ̸ = pR p , and the zero polynomial otherwise.We write the polynomial P R p ;I 1 ,...,I p as where the numbers e d (R p ; I 1 , . . ., I p ) are nonnegative integers called the mixed multiplicities of R p with respect to I 1 , . . ., I p .By summarizing the above results and discussion, it follows that the following expression and the zero polynomial otherwise.Furthermore, we can write P B(G) as where the numbers e d (B(G)) are nonnegative integers and given by So, the nonnegative integers e d (B(G)) gather the mixed multiplicities of the base points, and we call them the mixed multiplicities of the base locus of G.
We are now ready to state the main result of this subsection.
Theorem 3.8.Assume Setup 3.1 and suppose that G : is generically finite.Then, for all d = (d 1 , . . ., d p ) ∈ N p with |d| = r, the following statements hold: Proof.For any n = (n 1 , . . ., n p ) ∈ Z p + , we have the exact sequence of sheaves that yields the following exact sequence in cohomology This implies that for any i ⩾ 1 and n = (n 1 , . . ., n p ) ∈ Z p + with n i ≫ 0, the expression is a polynomial in n of degree strictly less than r.So, for any n = (n 1 , . . ., n p ) ∈ Z p + we have the equalities where is the polynomial function that encodes the mixed multiplicities of the saturated special fiber ring T = F(I 1 , . . ., I p ).Since F T (n) is a homogeneous polynomial of degree r in Q[n], it follows that the following function is also a homogeneous polynomial of degree r in Q[n].The fact that Q(n) ⩾ 0 for all n ∈ N r implies that the coefficients of Q(n) are all nonnegative.Therefore, the equality and Theorem 3.6 yield the general upper bounds p .This completes the proof of part (i).
Next, we assume that the base locus of G is zero-dimensional.Since dim(B(G)) = 0, we obtain the equalities So, combining the above expressions for Q(n), we obtain the following equality of polynomials And so, the result of part (ii) follows from Theorem 3.6.This completes the proof of the theorem.□ As a direct consequence of the above theorem, we obtain the following result.
Corollary 3.9.Assume Setup 3.1 and suppose that all the ideals I i are m-primary (i.e. the rational map G is actually a morphism G : Then, for all d = (d 1 , . . ., d p ) ∈ N p with |d| = r, we have the equality

LINEAR RATIONAL MAPS
In this section, we revisit the case of linear multiview varieties that has been extensively studied.Here, we use the following setup.
Setup 4.1.Assume Setup 3.1 and let δ i = 1 for each i.That is, the forms f i,j defining the rational map k are linear.In addition, we will assume that the field is algebraically closed.
We start by recalling that, in this setting, the mixed multiplicities of the special fiber ring are either zero or one.Proof.This result follows directly from Theorem 3.8(i).□ Our next observation is the following interesting corollary of Theorem 3.6.
Corollary 4.3.Assume Setup 4.1 and suppose that G : is generically finite.Then, the saturated special fiber ring and the special fiber ring coincide.Explicitly, for each (n 1 , . . ., n p ) ∈ N p , we have By assumption, all forms f i,j defining G are linear and so the G is birational.□ We now restrict our attention to the case where the base locus of G is zero-dimensional.For each 1 ⩽ i ⩽ p, the rational map G i : P r k P m i k is then given by (x 0 , . . ., x r ) → A i (x 0 , . . ., x r ) T for some matrix Under the assumption that dim(B(G)) = 0, we set-theoretically obtain B(G) = {Ker(A i ) | 1 ⩽ i ⩽ p} and we have the equivalent conditions dim(R/I i ) ⩽ 1 and rank(A i ) ⩾ r.We note that each ideal I i is linear, hence prime.
Remark 4.4.Suppose that r = 3 and m i = 2 for all i.The assumption that the base locus is zerodimensional is equivalent to the defining matrices A i of the rational map G i be of rank 3.In particular, this is the case studied in [1] which we generalize in Corollary 4. Proof.Fix a point p ∈ B(G) in the base locus.We write for all of the ideals which correspond to the same point in the base locus.We proceed by calculating the value of e d (R p ; I 1 , . . ., I p ).Note that for each 1 ⩽ j ⩽ p we have So, it follows that e d (R p ; I 1 , . . .,

CERTAIN SPECIAL FAMILIES OF RATIONAL MAPS
In this section, we consider some special families of rational maps for which we give exact formulas for the product of the degree of the rational map with the multidegrees of its image.We shall extend the computations obtained in [14,19] for perfect ideals of height two and Gorenstein ideals of height three.Throughout this section we continue using Setup 3.1.
We begin by studying the multidegrees of the graph To simplify notation, let Then, for any (d 0 , d) ∈ N p+1 with d 0 + |d| = r, we shall consider the multidegrees deg (Γ ) = e (d 0 , d), R(I 1 , . . ., I p ) .We proceed by relating the multidegrees of the graph Γ with the mixed multiplicities of F(I 1 , . . ., I p ).The following result provides an extension of [16,Theorem 5.4] to our setting.
and the fact that H i M (R(I 1 , . . ., I p )) 0,n = H i b (R(I 1 , . . ., I p )) 0,n = 0 for all n ≫ 0, we obtain the isomorphism H i m (R(I 1 , . . ., I p )) 0,n ∼ = H i N (R(I 1 , . . ., I p )) 0,n for all n ≫ 0. Let X = Proj R-gr (R(I 1 , . . ., I p )) be the projective scheme obtained by considering R(I) as a singlegraded k-algebra with the grading of R. By using the relations between sheaf and local cohomologies, see e.g.[39,Corollary 1.5] and [30, Appendix A4.1], we have that The multigraded Hilbert polynomial of R(I 1 , . . ., I p ) is given by , see e.g.[42,Lemma 4.3].By [20,Theorem 4.4], we have that H i m (R(I 1 , . . ., I p )) 0 is a finitely generated N p -graded module over F(I 1 , . . ., Hence, the fact that H Proof. Consider the N p+1 -graded k-algebra T (m | I 1 , . . ., I p ).Since the ideals I 1 , . . ., I p are equally generated in degrees δ 1 , . . ., δ p , by Nakayama's lemma we obtain for all (n 0 , n) = (n 0 , n 1 , . . ., n p ) ∈ N p+1 .This shows that T (m | I 1 , . . ., I p ) and R(I 1 , . . ., I p ) have the same Hilbert function, and so the result of the lemma follows.□ The following proposition deals with the process of cutting the graph Γ with general hyperplanes in the target space P. Note that this is similar to [16,Proposition 5.6], where the case of cutting Γ with general hyperplanes in the source P r k was treated.
Proposition 5.3.Assume Setup 3.1 with k being an infinite field.Fix Proof.By using [16,Lemma 3.7], we choose (α 0 , . . ., α m i ) in such a way that becomes a filter-regular element on R(I 1 , . . ., I p ) and on For any p ∈ Spec(R) \ V(I i ), the localization s ⊗ R R p of the surjection s becomes an isomorphism.Hence, there is some l > 0 such that I l i • Ker(s) = 0. Known dimension computations give us that dim (R(I 1 , . . ., I p )) = r + 1 + p and dim gr I i R(I 1 , . . ., I i , . . ., I p ) = r + p, see e.g.[38, §17.5].For any finitely generated N p+1 -graded A-module M, we set Supp ++ (M) := Supp(M) ∩ MultiProj(A).Since h is a filter-regular element on gr I i R(I 1 , . . ., I i , . . ., I p ) and on R(I 1 , . . ., I p ), [16, Lemma 3.9] and [39, Lemma 1.2] imply that dim Supp ++ (Ker(s)) ⩽ dim Supp ++ R(I 1 , . . ., Finally, the short exact sequence 0 → Ker(s) → R(I 1 ,...,I p ) h R(I 1 ,...,I p ) s − → R S (J 1 , . . ., J p ) → 0 and the additivity of mixed multiplicities yield deg and so the proof of the proposition is complete.□ The next lemma will allow us to simplify the families of rational maps we consider in this section.It shows that the mixed multiplicities of F(I 1 , . . ., I p ) depend only on a certain part of the data given and is similar to [19,Proposition 2.11].
Proof.(i) By the assumption of I 1 , . . ., I p−1 being m-primary ideals, we have p R p for all p ∈ Spec(R)\{m} and n 1 , . . ., n p ∈ N. We then obtain the equality It follows that Choose e > 0 such that H 1 m (R(I p )) k = 0 for all k ⩾ e.From the grading chosen for the Rees algebra, it follows that for all n ⩾ 0 and k ⩾ e.This implies that I n p nδ p +k = (I n p ) sat nδ p +k for all n ⩾ 0 and k ⩾ e.Let P(n) := P F(I 1 ,...,I p ) (n 1 , . . ., n p ) be the multigraded Hilbert polynomial of F(I 1 , . . ., 1 ⩾ e such that P(n 1 , . . ., n p ) equals the Hilbert function of F(I 1 , . . ., I p ) when n i ⩾ n ′ i .We can also assume that Q(c, n) equals the Hilbert function of Therefore, with the above choices in place, we obtain the following equalities for all n ⩾ n ′ p .As a consequence, we obtain e (0, . . ., 0, r), F(I 1 , . . ., Finally, [16, Theorem 5.4] (or Theorem 5.1 above) implies that e (0, r), R(I p ) = e F(I p ) , and so we are done.□ We are now ready to state our main result in this section.We say that an ideal I ⊂ R satisfies the condition G r+1 when µ(I p ) ⩽ dim(R p ) for all p ∈ V(I) ⊂ Spec(R) with ht(p) < r + 1, where µ(I p ) is the minimal number of generators of I p .The following theorem gives exact formulas for the multidegrees of the saturated special fiber ring e d, F(I 1 , . . ., I p ) for several families of rational maps.Theorem 5.5.Assume Setup 3.1 and suppose that the ideals I 1 , . . ., I p−1 are m-primary and that m p ⩾ r.

MONOMIAL RATIONAL MAPS
In this section, we study the case of a monomial rational map.We shall express the mixed multiplicities of the saturated special fiber ring in terms of mixed volumes of some naturally constructed polytopes.The following setup is used throughout this section.Setup 6.1.Assume Setup 3.1 and suppose that the polynomials f i,k ∈ I i are monomials.So, I 1 , . . ., I p ⊂ R are monomial ideals.
First, we need to recall some important results and fix our notation.Let K = (K 1 , . . ., K p ) be a sequence of convex bodies in R r .For any sequence λ = (λ 1 , . . ., λ p ) ∈ N p of nonnegative integers, we denote by λK the Minkowski sum λK := λ 1 K 1 + • • • + λ p K p and by K λ the multiset K λ := p i=1 λ i j=1 {K i } of λ i copies of K i for each 1 ⩽ i ⩽ p.Let us recall Minkowski's classic theorem, see e.g.[32,Theorem 3.2,page 116].Theorem 6.2 (Minkowski).Vol r (λK) is a homogeneous polynomial of degree r.
We write the polynomial Vol r (λK) as Vol r (λK) = where MV r (−) denotes the mixed volume.
For a monomial ideal J ∈ R with monomial generating set G(J), we define the following lattice convex polytope We denote by π : R r+1 ↠ R r the natural projection given by (x 0 , x 1 . . ., x r ) → (x 1 . . ., x r ).We have the following important result that expresses the mixed multiplicities of I 1 , . . ., I p with respect to m in terms of mixed volumes.It should be mentioned that this result was generalized in [17,18] for not-necessarily-Noetherian graded families of ideals.
The following theorem is the main result of this section.which we note has degree 1.Let I 1 = (x, y, z) and I 2 = (x 3 , yz 2 , xy 2 ) be the corresponding base ideals.Let π be the map that projects onto the x and y coordinates.In Figure 2 we depict the polytopes π(Γ (I 1 )) and π(Γ (I 2 )) along with their Minkowski sum.
The parametric curve is the image of the rational map G : P 1 k P 2 k given by G(s, t) = (a 0 , a 1 , a 2 ), restricted to a particular affine patch.
The concept of a µ-basis naturally extends to higher dimensions.For example, µ-bases for surfaces G : P 2 of the rational map is a local complete intersection.As a result, the formula for the degree is deg , where the µ-basis of (a, b, c, d) has degrees µ 1 , µ 2 , µ 3 .In more recent work, [14, Theorem A] gives a generalization of this result.It is shown that, for a rational map G : P r k P s k determined by a perfect base ideal of height two, the degree of the image times the degree of the map equals an elementary symmetric polynomial evaluated at the degrees of the µ-basis.The main tool used in the proof of this formula is the saturated special fiber ring, see Section 3.
The study of µ-bases has played an important role in a number of applications.As mentioned above, they appear in the implicitization problem of parametric surfaces and curves.But, they are also an important tool for detecting singularities and in studying the degree and birationality of rational maps.They are nowadays an essential tool in many developments in CAGD, see e.g.[11-13, 40, 41, 51, 52].
More generally, in a somehow parallel story, µ-bases form part of the general syzygy-based approach for studying rational maps.This method appears to have been originally initiated in [37].In this paper, Hulek, Katz and Schreyer give a characterization of when a map is a Cremona transformation in terms of properties of the syzygies.The use of syzygies in studying rational maps has become an active and particularly fruitful research area, see e.g.[3-5, 8, 14, 16, 19, 24-26, 29, 31, 34, 35, 43-46, 49, 50, 53].
In this paper, we further extend the syzygy-based approach (a.k.a µ-bases) by considering a family of rational maps which has, until this point, been mostly unobserved.We are interested in rational maps from a projective variety to a multiprojective variety, typically, a rational map of the form G : k .It should be mentioned that rational maps from a multiprojective variety to a projective variety (i.e. a rational map typically of the form H : P m 1 k × k • • • × k P m p k P r k ) are an important gadget in the field of Geometric Modeling, see e.g.[3,51,52].We define and use a multigraded generalization of the saturated special fiber ring that allows us to obtain important results regarding the (nonlinear) multiview variety given as the image of G.
Our work is motivated by the study of multiview varieties that were considered in [2] and in [1].The classical multiview variety is the image of a linear rational map G : k is called a (pinhole) camera and is given by a full-rank 3 × 4 matrix A i .This setup naturally arises in the study of computer vision, and the multiview variety is the closure of the image Y of G.In general, the defining ideal J of Y is a Cartwright-Sturmfels ideal, and so it has many desirable properties, see [21,22].The kernel of A i is a single point in P 3 k and the base locus of G is the collection of these points.In the case that these points are distinct, by [2, Theorem 3.7], we have an explicit description for the multigraded Hilbert function of J.This result follows from explicitly computing the generic initial ideal of the defining ideal J of Y.Our primary goal is to show that similar relationships hold even in the nonlinear case.In particular, we focus on the highest degree coefficients of the multigraded Hilbert polynomial of J, which are the multidegrees of the multiprojective variety Y.
To study the ideal of the nonlinear multiview variety, see Setup 3.1, it is not possible to take the same approach as [2] since the generic initial ideals are not radical.Instead, we are able to compute the multidegrees of the image of G using the saturated special fiber ring.We show in Theorem 3.6 that our multigraded version of the saturated special fiber ring maintains similar important properties (cf.[8,Theorem 2.4]).Similarly to [2, Theorem 3.7], we show in Theorem 3.8 that the multidegrees of Y are related to the mixed multiplicities of the base points.In Section 4, we recover several known results in the setting of the classical linear multiview varieties (see Theorem 4.2, Corollary 4.3, Corollary 4.5).In Section 5, we provide explicit formulas for several interesting families of rational maps.The techniques used in Section 5 convey the depth and scalability of the saturated special fiber ring to study rational maps.We also prove some technical results that might be interesting in their own right, see Theorem 5.1, Proposition 5.3 and Lemma 5.4.In Section 6, we focus on the case where the rational map G is given by monomials, i.e. the ideals I 1 , . . ., I p are monomial ideals.Using convex geometry, we relate the multidegrees of the image with the mixed volumes of the associated Newton polytopes of the ideals I i after dehomogenizing.
Future directions.In this paper, we have computed for large families of rational maps, the multidegrees of their image.Following this, it is natural to ask how we may compute the other coefficients of the multigraded Hilbert polynomial.In particular, we ask whether these coefficients can be determined from the points in the base locus analogously to Theorem 3.8.Of course, it is also quite desirable to study other families of nonlinear multiview varieties.And so, we ask for which families of multiview varieties is possible to find explicit formulas for their multidegrees.

Notation 3 . 2 .
Let M be an N p+1 -graded module over A. We denote by [M] c the "one-sided" R-graded part [M] c = n∈Z p [M] c,n .Then, [M] c has a natural N p -graded structure, and its n-th graded part is given by [M] c n = [M] c,n where n ∈ N p .
(i) For all d ∈ N p with |d| = r, we have the equality e d, F(I 1 , . . ., I p ) = deg(G) • deg d P (Y).

Theorem 4 . 2 (
[47, Theorem 1.1]  and[23, Theorem 3.9]).Assume Setup 4.1 and suppose that the mapG : P r k P m 1 k × k • • • × k P m p kis generically finite.Then, the closure Y of the image of G is multiplicityfree.That is, deg d P (Y) ∈ {0, 1} for all d ∈ N p with |d| = r.

5 . 1 . 4 . 5 .
By applying Theorem 3.8, we can give an explicit description of the degrees d, where |d| = r, such that deg d P (Y) = Corollary Assume Setup 4.1 and suppose that dim(B(G)) = 0. Let d = (d 1 , . . ., d p ) ∈ N p with |d| = r.Then deg d P (Y) = 1 if and only if for each point p ∈ B(G) there exists 1 ⩽ i ⩽ p such that I i ̸ = p and d i ⩾ 1.
see [38, §17.4].By Theorem 3.8, we have 1 = deg d P (Y) deg(G) + e d (B(G)).The mixed multiplicities of the base locus are given by [κ(p) : k]e d (R p ; I 1 , . . ., I p ) where the sum is taken over the set of primes p ∈ B(G).Since, k is algebraically closed, for all p ∈ B(G), we have [κ(p) : k] = 1.Finally, it follows that deg d P (Y) = 1 if and only if for each point p ∈ B(G) there exists some 1 ⩽ i ⩽ p such that I i ̸ = p and d i ⩾ 1.□
(a) I p is a Gorenstein ideal of height three.(b) Every nonzero entry of an alternating minimal presentation matrix of I p has degree D ⩾ 1. (c) I p satisfies the condition G r+1 .Then, for all d = (d 1 , . . ., d p ) ∈ N p with |d| = r, we have deg(G) • deg d P

FIGURE 1 .Example 6 . 6 .
FIGURE 1.The polytopes associated to the ideals in Example 6.5 and their projections.

k P 3 k
: (s, t) → (a, b, c, d) are studied in [27, Section 5].The µ-basis allows us to determine properties about the parametric surface, i.e. the image of G.Under particular assumptions, Cox computes in [27, Proposition 5.3] that the degree of the image of G can be determined completely in terms of the degrees of a µ-basis.Explicitly, Cox requires that deg(G) = 1 and that the base locus V(a, b, c, d) ⊆ P 2 k [22, p .Proof.By Theorem 4.2 we have that F(I 1 , ..., I p ) is multiplicity-free.Hence, by a theorem of Brion, see[22, Theorem 1.11] and [7, Theorem 2], the special fiber ring F(I 1 , . . ., I p ) is integrally closed.We may therefore apply Theorem 3.6(ii) to deduce that F(I 1 , . . ., I p ) = F(I 1 , . . ., I p ) if and only if G is birational.