A Lower Bound on Determinantal Complexity
aa r X i v : . [ c s . CC ] S e p A Lower Bound on Determinantal Complexity
Mrinal Kumar ∗ Ben Lee Volk † Abstract
The determinantal complexity of a polynomial P ∈ F [ x , . . . , x n ] over a field F is the di-mension of the smallest matrix M whose entries are affine functions in F [ x , . . . , x n ] such that P = Det ( M ) . We prove that the determinantal complexity of the polynomial P ni =1 x ni is atleast . n − .For every n -variate polynomial of degree d , the determinantal complexity is trivially at least d , and it is a long standing open problem to prove a lower bound which is super linear in max { n, d } . Our result is the first lower bound for any explicit polynomial which is bigger by aconstant factor than max { n, d } , and improves upon the prior best bound of n + 1 , proved byAlper, Bogart and Velasco [ABV17] for the same polynomial. The determinantal complexity of a polynomial f ∈ F [ x , . . . , x n ] , denoted dc ( f ) , is the minimalinteger m such that there exists an affine map L : F n → F m × m such that f ( x ) = Det ( L ( x )) ,where for every square matrix M , Det ( M ) denotes the determinant of M .This notion was first implicitly defined by Valiant [Val79], and it is tightly related to the VP vs. VNP problem, the algebraic analog of the P vs. NP problem. The essence of the VP vs. VNP problem is showing that some explicit polynomials are hard to compute. By defining naturalnotions of reductions and completeness, Valiant showed that this problem is in fact equivalentto showing that, for fields of characteristic different than two, the determinantal complexity ofthe permanent polynomial,
Perm n ( X ) = X σ ∈ S n n Y i =1 x i,σ ( i ) , doesn’t grow like a polynomial function in n. This fact is a consequence of the completeness property of the determinant: Valiant showedthat if f has an algebraic formula of size s , then the determinantal complexity of f is at most s . This remains true even if f has an algebraic branching program (ABP) of size s : ABPs are anatural and more powerful model of computation than formulas. We refer to [SY10] and [Sap15]for more background on algebraic complexity theory and for proofs of these statements.Thus, Valiant also established en passant the non-obvious fact that the determinantal com-plexity of every polynomial is finite, and it’s at most roughly (cid:0) n + dn (cid:1) for every n -variate polynomial ∗ Department of Computer Science & Engineering, IIT Bombay. Email: [email protected] † Department of Computer Science, University of Texas at Austin, USA. Part of this work done while at the Centerfor the Mathematics of Information, California Institute of Technology, USA. Email: [email protected] Strictly speaking, the VP vs. VNP question is equivalent to showing that the determinantal complexity of the
Perm n is at least n ω (log n ) , but we skip over this fine grained detail for now. f degree d . Standard counting arguments also show that this estimate is close to being tightfor almost every such polynomial.The benefit of this reformulation of the VP vs. VNP problem is that it appears to stripaway altogether the notion of “computation”: indeed, this problem can be stated without evendefining a computational model in any standard sense of the word, and thus it can potentiallybe proved without having to argue about the topology or structure of every possible arithmeticcomputation.In practice, however, proving lower bounds on determinantal complexity is (unsurprisingly)difficult. Currently, for n -variate polynomials, there are no known lower bounds which aresuper-linear in n (see Section 1.2 for more details on previous work). Due to the completenessproperty mentioned above, a lower bound of s on the determinantal complexity of f will implythe same lower bound for algebraic formulas and even algebraic branching programs. However,super-linear lower bounds for formulas are well-known for decades [Kal85], and super-linearlower bounds for ABPs were recently established in [CKSV19], so there doesn’t seem to be anymajor complexity-theoretic barrier for proving such lower bounds for determinantal complexity:the main obstacle is seemingly lack of techniques for reasoning about computations using deter-minants, and hence it is important to study this model and to develop techniques to understandit and to prove lower bounds, for the permanent as well as for other explicit polynomials.Even for the purpose of separating VP and VNP , one need not necessarily prove a lowerbound on the determinantal complexity of the permanent; the same conclusion will hold if thelower bound on determinantal complexity is shown for any “explicit” polynomial (formally, inthe class
VNP , which we don’t define here) in lieu of the permanent.Before we describe the previous work concerning determinantal complexity, we provide a briefremark about the notion of a “trivial” lower bound in this context which is worth rememberingwhen evaluating the previous results (and our result). Unlike most standard computationalmodels, observe that for an n -variate polynomial of degree d , even a lower bound of n is non-trivial for determinantal complexity. This is because every coordinate of the affine map L candepend on all n variables. Nevertheless, since the determinant of an m × m matrix is a degree m polynomial, and thus Det ( L ( x )) is a degree m polynomial for every affine map L , the degree d is a trivial lower bound on the determinantal complexity of f . Therefore, it is natural to considerpolynomial families in which d ≤ n or alternatively to hope to prove lower bounds stronger than max { n, d } . The early work on determinantal complexity mostly focused on proving lower bounds for thepermanent. Recall that the n × n permanent, Perm n , is a degree n polynomial, so the triviallower bound is dc ( Perm n ) ≥ n . Since over characteristic 2 the permanent and determinantcoincide, the results described here hold for characteristic not equal to 2.Already in 1913, Szegő [Sze13], answering a question of Pólya [Pól13], showed that there’sno way to generalize the × identity Perm (cid:18) x , x , x , x , (cid:19) = Det (cid:18) x , x , − x , x , (cid:19) by affixing ± signs to an n × n matrix of variables for n ≥ .Marcus and Minc [MM61] strengthened this result by showing that for every n , dc ( Perm n ) >n . Subsequent work by von zur Gathen [vzG87], Babai and Seress (see [vzG87]), Cai [Cai90]and Meshulam [Mes89] obtained the slightly stronger lower bound dc ( Perm n ) ≥ √ n .Mignon and Ressayre [MR04] greatly improved the lower bound by proving dc ( Perm n ) ≥ n / , over the complex numbers. Cai, Chen and Li [CCL10] extended this lower bound to fieldsof positive characteristic different than two, and Landsberg, Manivel and Ressayre [LMR13] xtended this result to the border version of determinantal complexity, that is, they showedthat the permanent is not even in the closure of polynomials with determinantal complexity lessthan n / . Finally, Yabe [Yab15] obtained an improved lower bound of ( n − + 1 over thereal numbers.However, while these lower bounds are quadratic in the degree, Perm n is a polynomial with n many variables, and notably none of these lower bounds is larger than the number of variables.In particular, these results don’t even recover a weak form of the n formula lower bound ofKalorkoti for Perm n [Kal85].Landsberg and Ressayre [LR17] considered determinantal representations that respect cer-tain symmetries (which they called equivariant determinantal complexity and denoted edc ), andproved that edc ( Perm n ) is exponential in n . It’s unclear how stringent the symmetry require-ment is; Ladnsberg and Ressayre put forward the ambitious conjecture that edc and dc arepolynomially related, which, if true, would imply VP = VNP . To the best of our knowledge, thisconjecture remains open, but it’s worth mentioning that in the context of regular determinantalcomplexity , another notion defined and studied by [LR17], it can be shown unconditionally thatrequiring symmetry may result in a super-polynomial blow-up. The question of lower bounds for other explicit polynomial was also considered: Mignon andRessayre [MR04] proved that the determinantal complexity of quadratic polynomials of rank r is exactly ⌈ ( r + 1) / ⌉ (this, of course, cannot give a lower bound beyond ⌈ ( n + 1) / ⌉ ). Chen,Kayal and Wigderson [CKW11] observed that the technique of Mignon and Ressayre implies an n/ lower bound on the determinantal complexity of the elementary symmetric polynomial ofdegree 2, P ≤ i Theorem 1.1. For every natural number n ≥ , the determinantal complexity of the polynomial P ni =1 x ni over the field of complex numbers is at least . n − . Although for simplicity we state our results for the complex numbers, all the results in thispaper also hold for algebraically closed fields of positive characteristic p , as long as p doesn’tdivide n . This assumption is not only an artifact of the proof. For example, when n = p k , and The regular determinantal complexity of a polynomial f is the smallest integer m such that there is an m × m matrix M ( x ) of linear forms such that f ( x ) = Det ( M ( x )) and rank ( M ( )) = m − . This definition is motivatedby the fact that for the permanent and for many other polynomials, every determinantal representation is regular,without loss of generality (see Lemma 3.5). The regular determinantal complexity of every polynomial f with size s formula is at most s + 1 , as can be witnessed by the reductions in Exercise 13.2 of [CKW11] or Section 3.3.1 of[Sap15], which satisfy the regularity property. Since the determinant polynomial Det n has a formula of size n O (log n ) ,this implies a similar upper bound on its regular determinantal complexity, while Landsberg and Ressayre [LR17]proved that the equivariant regular determinantal complexity of Det n is exponential. ver characteristic p , p k X i =1 x p k i = p k X i =1 x i p k has determinantal complexity at most n = p k ; it is also a polynomial of degree n , so its deter-minantal complexity is at least, and hence equals, n .As discussed in Section 1.2, this is the first non-trivial lower bound of the form (1 + ǫ ) n , forany ǫ > for any explicit n variate polynomial family, and improves the previous best boundof n + 1 by Alper, Bogart and Velasco [ABV17] by a constant factor.This result, of course, is not fully satisfactory. The best upper bound we’re aware of for dc ( P ni =1 x ni ) is O ( n ) , which follows from converting the natural algebraic formula or ABPcomputing this polynomial to a determinantal expression. We suspect that the true complexitymight be Ω( n ) or at the very least ω ( n ) .Quantitatively, the situation here is somewhat similar to the case of lower bounds on therank of 3-dimensional tensors, where the best lower bounds are only a constant factor awayfrom the trivial lower bound, and proving super-linear lower bounds remains a challenging openproblem (cf. [AFT11, BD80, Blä99, Shp01], among others).We now give an outline of the main ideas in our proof. Let M ∈ F [ x , x , . . . , x n ] m × m matrix of affine functions such that P ni =1 x ni = Det ( M ( x )) .Theorem 1.1 shows a lower bound of . n − on m . There are essentially three main ingredientsto the proof of Theorem 1.1, and we now discuss them in some more detail. Converting the matrix M into a normal form Let M ∈ F m × m be the constant part of the matrix M , i.e. M = M ( ) . As a first step ofour proof, we show (in Lemma 3.2) that without loss of generality, M can be assumed to be adiagonal matrix of rank equal to m − . We a say that a matrix M is in normal form if it hasthis additional structure.Is is quite easy to observe that the rank of M is at most m − . However, for technicalreasons, we actually need the lower bound on the rank as well, and this fact is a consequence ofcomparing the dimensions (as algebraic varieties) of the singular locus (which is just the the set ofzeroes of a polynomial of multiplicity at least two) of the determinant and that of the polynomial P ni =1 x ni . Observations of this nature have been used in the context of determinantal complexitylower bounds before, and indeed, we crucially rely on a well known lemma of von zur Gathen(see Fact 3.4) for the proof. The details can be found in Section 3.1. Determinantal complexity of higher degree polynomial maps As the key ingredient of our proof, we show that for any matrix M ( x ) ∈ F [ x ] m × m where theentries of M are polynomials of degree at most n − and M is in normal form, if Det ( M ( x )) = P ni =1 x ni , then m ≥ n/ . Moreover, roughly the same lower bound continues to hold as long as det( M ) = ( P ni =1 x ni ) (1 + Q ) for any polynomial Q , with Q ( ) = 0 .Thus, this is a significant generalization of the n/ lower bound on the standard notiondeterminantal complexity (where the entries of M are affine functions) of P ni =1 x ni as shown in[Kum19]: this shows that roughly the same lower bound continues to hold even when the entries This means that the degree of the polynomials is at most the number of variables. f the matrix are arbitrary polynomials of degree as high as n − and the determinant of thematrix equals an arbitrary multiple of P ni =1 x ni with a non-zero constant term.The proof of the lemma relies on the observation that the polynomial P ni =1 x ni does notvanish with multiplicity at least two very often. This seemingly simple observation has beenpreviously used in the context of lower bounds on algebraic branching programs computing thispolynomial [Kum19, CKSV19] in a crucial way. See Section 3.2 for further details. Trading dimension of the matrix for degree As the final ingredient of our proof, we use a well known property of determinants (Lemma 3.8)to show that if there is an m × m matrix M whose entries are affine functions and Det ( M ) = P ni =1 x ni , then there is an ( m − n + 2) × ( m − n + 2) matrix N whose entries are polynomialsof degree at most n − and Det ( N ) = ( P ni =1 x ni )(1 + Q ) for a polynomial Q which vanishes atzero. Moreover, if the matrix M is in normal form, then the matrix N continues to be in normalform.Thus, we are in a setup where we can invoke the lower bound in Lemma 3.9 discussed earlierand we get that the dimension of N which equals m − n + 2 must be at least n/ − , therebyimplying that m is at least . n − . The details of this step can be found in Section 3.3. In this paper F always denotes an algebraically closed field. We use x to denote a tuple of n variables x , . . . , x n , where n is understood from the context (or is otherwise explicitly men-tioned).We consider polynomial maps M : F n → F m × m given by m polynomials ( M i,j ) i,j ∈ [ m ] . Thesame object can be thought of as a matrix of polynomials M ( x ) ∈ F [ x ] m × m and we use bothpoints of view interchangeably. The degree of M is the maximum degree of its coordinates, i.e., deg M = max i,j deg M i,j .Each M ( x ) ∈ F [ x ] m × m can be uniquely written as M ( x ) = M ′ ( x ) + M , where M ∈ F m × m and in all m coordinates of M ′ , the constant term is zero. We then call M the constantpart of the map. A polynomial in which the constant term is zero is called constant free , anda polynomial map is called constant free if all of its coordinates are constant free, i.e., in theabove decomposition, M = 0 .We denote the determinant polynomial by Det . In cases where it is important to emphasizethe dimension of the matrices in question we write it in the subscript, so for example the m × m determinant polynomial is denoted by Det m . Determinantal Complexity We now formally define the notion of determinantal complexity, which is the focus of this paper. Definition 2.1 (Determinantal Complexity) . The determinantal complexity of a polynomial P ∈ F [ x ] is defined as the minimum m ∈ N such that there is a m × m matrix M ∈ F [ x ] whoseentries are polynomials of degree at most one such that P = Det ( M ) . ♦ Remark 2.2. The above definition naturally generalizes to a family of polynomials in the fol-lowing sense. A family { P n } n ∈ N of polynomials is said to have determinantal complexity at most f ( n ) : N → N if there exists an n ∈ N , such that for every n ≥ n , the determinantal complexityof P n is at most f ( n ) . ♦ A lower bound on determinantal complexity This section will be devoted for a proof of Theorem 1.1. We begin with the following lemma,which was instrumental in the recent proofs of lower bounds for algebraic formulas and algebraicbranching programs. Lemma 3.1 ([CKSV19, Kum19]) . Let d ≥ be a natural number. Let P , P , . . . , P t , Q , . . . , Q t ,R ∈ C [ x ] be polynomials such that deg( R ) < d , P , . . . , P t , Q , . . . , Q t have a common zero and n X i =1 x di = R + t X j =1 P j ( x ) Q j ( x ) . Then, t ≥ n/ . We now show that without loss of generality, the constant part of every polynomial map M such that P ni =1 x di = Det m ( M ( x )) has a very special form: is it an m × m diagonal matrix with in the (1 , coordinate and in all diagonal entries. M to a normal form This claim is not entirely new and very similar statements were proved, for example, in [MR04,ABV17]. For completeness, and since the exact statement we need is slightly more general, weprovide a proof. Lemma 3.2. Let d ≥ be a natural number and let M ( x ) ∈ F [ x ] m × m be a polynomial mapsuch that Det m ( M ( x )) = n X i =1 x di . Then, there exists a matrix ˜ M ( x ) ∈ F [ x ] m × m with deg( ˜ M ) ≤ deg( M ) , Det m ( ˜ M ( x )) = n X i =1 x di , and the constant part of ˜ M is a diagonal m × m matrix ˜ M such that ( ˜ M ) , = 0 and ( ˜ M ) i,i = 1 ,for ≤ i ≤ m . To prove Lemma 3.2 we require a few preliminaries. We begin with the definition of asingular locus of a polynomial (or a hypersurface). Definition 3.3. Let f ∈ F [ x ] be a polynomial. The singular locus of f , denoted Sing( f ) , is thevariety defined by Sing( f ) = (cid:26) a : ∂f∂x i ( a ) = 0 , ≤ i ≤ n (cid:27) . ♦ The singular locus of the determinant was studied by von zur Gathen, who proved thefollowing fact. Fact 3.4 ([vzG87]) . Let F be an algebraically closed field and let Det m denote the m × m determinant polynomial. Then Sing( Det m ) ⊆ F m × m is precisely the set of matrices of rank atmost m − , and dim Sing( Det m ) = m − . The following is a slight generalization of a lemma of von zur Gathen (cf. also [ABV17]). Lemma 3.5. Let f ∈ F [ x ] be a polynomial, and let M : F n → F m × m be a polynomial mapsuch that f ( x ) = Det m ( M ( x )) . Suppose further that dim(Sing( f )) < n − . Then Im( M ) ∩ Sing( Det m ) = ∅ . Furthermore, all matrices in Im( M ) have rank at least m − . roof. Let y i,j denote the coordinates of F m × m and write M = ( M i,j ) i,j ∈ [ m ] . Using the chainrule, we compute ∂f∂x k = X i,j ∈ [ m ] ∂ Det m ∂y i,j ( M ( x )) · ∂M i,j ∂x k ( x ) , k ∈ [ n ] . (3.6)Suppose A ∈ Im( M ) ∩ Sing( Det m ) , and let B be such that A = M ( B ) . By definition of Sing( Det m ) , ∂ Det m ∂y i,j ( M ( B )) = 0 for all i, j ∈ [ m ] , and by (3.6) we get that B ∈ Sing( f ) . Thus M − (Sing( Det m )) ⊆ Sing( f ) , and dim( M − (Sing( Det m ))) ≤ dim Sing( f ) < n − . On the otherhand, using a standard lower bound on the dimension of pre-images of polynomial maps (seeTheorem 17.24 of [Har95]), if Im( M ) and Sing( Det m ) aren’t disjoint, dim( M − (Sing( Det m ))) ≥ n + ( m − − m = n − . This contradiction implies that Im( M ) ∩ Sing( Det m ) = ∅ . The “furthermore” part of the theoremfollows from Fact 3.4.We will also need the following easy fact which shows that P ni =1 x di satisfies that assumptionof Lemma 3.5. Fact 3.7 ([Kum19, CKSV19]) . For every d ≥ , dim(Sing( P ni =1 x di )) = 0 . We are now ready to prove Lemma 3.2. Proof of Lemma 3.2. Let f = P ni =1 x di and let M : F n → F m × m be a polynomial map such that f ( x ) = Det m ( M ( x )) , and write M = M ′ + M where M is the constant part of M .First, observe that f ( ) = Det m ( M ( )) = Det m ( M ) , which implies that rank ( M ) < m . By Lemma 3.5 and Fact 3.7, we also know that rank ( M ) = rank ( M ( )) ≥ m − , so rank ( M ) = m − .By performing Gaussian elimination on the rows and on the columns, we can find two m × m matrices G , G such that det( G i ) = ± for i = 1 , and N := G M G is a diagonal matrixsuch that ( N ) , = 0 and ( N ) i,i = 0 for ≤ i ≤ m .Now define a diagonal m × m matrix ∆ such that ∆ i,i = 1 / ( N ) i,i for ≤ i ≤ m , and ∆ , = Det ( G ) · Det ( G ) · m Y i =2 ( N ) i,i . It readily follows that Det (∆) = Det ( G ) · Det ( G ) , and that ˜ M := ( G M G )∆ is a diagonalmatrix such that ( ˜ M ) , = 0 and ( ˜ M ) i,i = 1 for all ≤ i ≤ m .Finally, define ˜ M = G M G ∆ . We verify that indeed Det ( ˜ M ( x )) = Det ( G ) · Det ( M ( x )) · Det ( G ) · Det (∆)= Det ( M ( x )) · ( Det ( G ) · Det ( G )) = Det ( M ( x )) = f ( x ) . We also have that ˜ M = G ( M ′ + M ) G ∆ = G M ′ G D + G M G ∆ = G M ′ G ∆ + ˜ M . Since G , G , ∆ ∈ F m × m , it also holds that ˜ M ′ := G M ′ G ∆ is a matrix of constant-freepolynomials, and that deg ˜ M ≤ deg M .We will also use the following simple and well known property of the determinant of a blockmatrix. emma 3.8. Let M ∈ F m × m be a matrix, and let A ∈ F t × t , B ∈ F t × m − t , C ∈ F m − t × t , D ∈ F m − t × m − t be its submatrices as follows: M = (cid:18) A BC D (cid:19) If D is invertible, then Det ( M ) = Det ( A − BD − C ) · Det ( D ) . Proof. Follows directly from the decomposition (cid:18) A BC D (cid:19) = (cid:18) A − BD − C BD − I m − t (cid:19) · (cid:18) I t C D (cid:19) and the multiplicativity of the determinant. In the following lemma we prove a lower bound of n/ on the determinantal complexity in a moregeneral model than the standard model. This is a generalization with respect to two properties.First, the entries of the matrix are no longer constrained to be polynomials of degree at most , and can have degree as high as d − , while computing the degree d polynomial (cid:0)P ni =1 x di (cid:1) .Moreover, the determinant of the matrix M does not even have to compute the candidate hardpolynomial (cid:0)P ni =1 x di (cid:1) exactly. It suffices if the determinant is equal to a polynomial of the form (cid:0)P ni =1 x di (cid:1) · ( β + Q ) where β is a non-zero field constant and Q is an arbitrary polynomial (ofpotentially very high degree!) which is constant free, i.e. Q ( ) = 0 . Lemma 3.9. Let d ≥ be a natural number and let M ( x ) ∈ F [ x ] m × m such that deg( M ) ≤ d − ,and the constant part of M is a diagonal matrix M such that ( M ) , = 0 and ( M ) i,i = 1 for ≤ i ≤ m . Suppose that Det ( M ) = n X i =1 x di ! · ( β + Q ) , where β ∈ F is non-zero and Q is a constant free polynomial. Then m ≥ n/ − .Proof. Using the Laplace expansion of Det ( M ) along the first row, we get Det ( M ) = m X j =1 ( − ( j +1) M ,j · Det ( N ,j ) , where N i,j is the submatrix of M obtained by deleting the i -th row and the j -th column. Forevery j ∈ [ m ] , j > , we claim that Det ( N ,j ) is a constant free polynomial, i.e. Det ( N ,j )( ) = Det ( N ,j ( )) = 0 . To see this, we observe that for every j ∈ [ m ] \ { } , N ,j ( ) is a ( m − × ( m − matrix, whichhas at most m − non-zero entries. This follows since M has at most m − non-zero entriesand in obtaining N ,j from M , we drop the entry M j,j , which is one of the ( m − entries of M with a non-zero constant term, and hence one of the ( m − non-zero entries of M . However,we note that N , ( ) is the ( m − × ( m − identity matrix, so the constant term of Det ( N , ) is , and we write Det ( N , ) = 1 + P ( x ) where P is constant free. Therefore, we have n X i =1 x di ! · ( β + Q ) = Det ( M ) = M , (1 + P ) + m X j =2 ( − ( j +1) M ,j · Det ( N ,j ) n other words, n X i =1 x di ! · ( β + Q ) = Det ( M ) = M , + M , · P + m X j =2 ( − ( j +1) M ,j · Det ( N ,j ) Slightly rearranging (and using β = 0 ), we get n X i =1 x di = 1 β − n X i =1 x di ! · Q + M , + M , · P + m X j =2 ( − ( j +1) M ,j · Det ( N ,j ) Since, deg( M , ) < d and M , , P, M , , Det ( N , ) , . . . , M ,k , Det ( N ,k ) , Q are all constant free(and hence share a common zero, namely ), we have from Lemma 3.1 that m ≥ n/ − . We are now ready to complete the proof of Theorem 1.1. Proof of Theorem 1.1. Let M be a m × m matrix with deg( M ) ≤ such that n X i =1 x ni = Det ( M ) . From Lemma 3.2, we can assume without loss of generality that the constant part M of M is adiagonal matrix such that ( M ) , = 0 and ( M ) i,i = 1 for ≤ i ≤ m . In particular, all the offdiagonal entries of M and M , are homogeneous linear forms or zero, and M j,j = 0 for j > .Observe that for every t ≤ m − , the principle minor D t of M which is obtained by deletingthe first m − t rows and columns of M are all invertible over the field of rational functions F ( x ) .To see this, observe that the matrix D t ( ) is the identity matrix, which implies that Det ( D t ) is a non-zero polynomial. Moreover, since every entry of M has degree at most , and Det ( M ) has degree n , we know that m ≥ n . So, we conclude that the principle minor D := D ( n − of M is invertible over F ( x ) . Thus, if B and C are respectively the submatrices of M defined as M = (cid:18) A BC D (cid:19) then by Lemma 3.8 we have Det ( M ) = Det ( A − BD − C ) · Det ( D ) . (3.10)Since D − = adj( D ) / det( D ) , where adj( D ) is the adjugate matrix of D , the entries of D − canbe written as as a ratio of two polynomials, where the numerator has degree at most n − andthe denominator, which is equal to Det ( D ) , has degree at most n − . Moreover, as discussedearlier in the proof, the constant part of D is the identity matrix, so there is a constant freepolynomial Q ∈ F [ x ] such that Det ( D ) = 1 + Q . Thus, every entry of the ( m − n + 2) × ( m − n + 2) matrix A − BD − C can be written as aratio of two polynomials with the numerator being a polynomial of degree at most n − andthe denominator being equal to Det ( D ) = 1 + Q . Therefore, by clearing the denominators andusing (3.10), we get that Det ( M ) · (1 + Q ) m − n +2 = Det ( N ) · (1 + Q ) , here N is the matrix with polynomial entries of degree at most n − obtained by multiplyingevery entry of A − BD − C by Q . Simplifying further, we get n X i =1 x ni ! · (1 + Q ) m − n +1 = Det ( M ) · (1 + Q ) m − n +1 = Det ( N ) . We are almost ready to invoke Lemma 3.9 to obtain a lower bound on the size of N (and hence M ), but to do that we need to ensure that the constant part of N , N , is a diagonal matrixwith ( N ) , = 0 and ( N ) i,i = 1 for ≤ i ≤ m − n + 2 . We now verify that this is indeed thecase.Recall that by the structure of the constant part M of M , all the entries of B and C and the (1 , entry of A are constant free, and the constant term of A i,i is for ≤ i ≤ m − n + 2 . Thus,every entry of the matrix BD − C is a rational function with a constant free numerator, andhence all the off-diagonal entries in A − BD − C as well as its (1 , entry are rational functionswith a constant free numerator. Moreover, the denominator of all the entries (cid:0) A − BD − C (cid:1) equals Det ( D ) = 1 + Q , for a constant free polynomial Q . So, expressing each element of A − BD − C as a quotient of polynomials, the constant term of each numerator on the diagonalis except for the (1 , entry, which has a constant free numerator. Finally, observe thateliminating the denominator of the entries of (cid:0) A − BD − C (cid:1) by multiplying every entry by (1 + Q ) gives us the matrix N .Thus the matrix N satisfies the hypothesis of Lemma 3.9, and hence ( m − n + 2) ≥ n/ − .This gives us m ≥ . n − and completes the proof of Theorem 1.1. Acknowledgment Mrinal thanks Ramprasad Saptharishi for various discussions on determinantal complexity overthe years, and in particular for explaining the proof of the result of Mignon and Ressayre tohim. References [ABV17] Jarod Alper, Tristram Bogart, and Mauricio Velasco. A Lower Bound for the De-terminantal Complexity of a Hypersurface. Found. Comput. Math. , 17(3):829–836,2017.[AFT11] Boris Alexeev, Michael A. Forbes, and Jacob Tsimerman. Tensor Rank: Some Lowerand Upper Bounds. 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