Computing all Affine Solution Sets of Binomial Systems
aa r X i v : . [ c s . S C ] M a y COMPUTING ALL AFFINE SOLUTION SETS OF BINOMIAL SYSTEMS(EXTENDED ABSTRACT)
DANKO ADROVIC AND JAN VERSCHELDE
Abstract.
To compute solutions of sparse polynomial systems efficiently we have to ex-ploit the structure of their Newton polytopes. While the application of polyhedral methodsnaturally excludes solutions with zero components, an irreducible decomposition of a va-riety is typically understood in affine space, including also those components with zerocoordinates. For the problem of computing solution sets in the intersection of some co-ordinate planes, the direct application of a polyhedral method fails, because the originalfacial structure of the Newton polytopes may alter completely when selected variablesbecome zero. Our new proposed method enumerates all factors contributing to a general-ized permanent and toric solutions as a special case of this enumeration. For benchmarkproblems such as the adjacent 2-by-2 minors of a general matrix, our methods scale muchbetter than the witness set representations of numerical algebraic geometry. Introduction
Our investigation in [3] starts with the sparsest kind of polynomial systems: those withexactly two monomials with nonzero coefficients in every equation. This sparsest type ofsystems is called binomial . Software implementations of primary decompositions of binomialideals [9] are described in [6] and [19]. Recent algebraic algorithms are developed in [15]and [18]. The complexity of counting the total number of affine solutions of a system of n binomials in n variables was shown as Monomial Maps representing Affine Solution Sets
Solution sets of binomial systems can be described as monomial maps, obtained via uni-modular coordinate transformations [1], see also [10] and [14]. Note that some sparse poly-nomial systems such as the cyclic n -roots problems have monomial maps as solution sets [2]. Definition 2.1. A monomial map of a d -dimensional solution set in C n is(1) x k = c k t v ,k t v ,k · · · t v d,k d , c k ∈ C , v i,k ∈ Z , for i = 1 , , . . . , d and k = 1 , , . . . , n .For a toric solution, all coefficients c k in the monomial map (1) are nonzero. For an affinesolution set, several coordinates may be zero. When setting variables to zero, it may happenthat all constraints on some other variables vanish, then we say that those variables are free ,while others are still linked to a toric solution of a subset of the original equations. Date : 29 April 2014. A Generalized Permanent
To enumerate all choices of variables to be set to zero, we use the matrix of exponents ofthe monomials to define a bipartite graph between monomials and variables.
Definition 3.1.
Let f ( x ) = be a system. We collect all monomials x a that occur in f along the rows of the matrix, yielding the incidence matrix (2) M f [ x a , x k ] = (cid:26) a k >
00 if a k = 0 . Variables which occur anywhere with a negative exponent are dropped.
Example 3.2.
For all adjacent minors of a 2-by-3 matrix, the incidence matrix is(3) M f = x x x x x x x x x x x x x x for the system defined by f = ( f , f ) with f = x x − x x and f = x x − x x .For this example, the rows of M f equal the exponents of the monomials. We select x and x as variables to be set to zero, as overlapping columns x with x gives all ones. Proposition 3.3.
Let S be a subset of variables such that for all x a occurring in f ( x ) = : M [ x a , x k ] = 1 , for x k ∈ S , then setting all x k ∈ S to zero makes all polynomials of f vanish.Proof. M [ x a , x k ] = 1 means: x k = 0 ⇒ x a = 0 . If the selection of the variables in S is suchthat all monomials in the system have at least one variable appearing with positive power,then setting all variables in S to zero makes all monomials in the system vanish. (cid:3) Enumerating all subsets of variables so that f vanishes when all variables in a subset areset to zero is similar to a row expansion algorithm on M f for a permanent: Algorithm 3.4 (recursive subset enumeration via row expansion of permanent) . Input: M f is the incidence matrix of f ( x ) = ;index of the current row in M f ; and S is the current selection of variables.Output: all S that make the entire f vanish.if M [ x a , x k ] = 1 for some x k ∈ S then print S if x a is at the last row of M f or else go to the next rowelse for all k : M [ x a , x k ] = 1 do S := S ∪ { x k } if x a is at the last row of M f then print S else go to the next row S := S \ { x k } Greedy enumeration strategies can be applied in the algorithm above. The enumerationmay generate subsets of variables that lead to affine monomial maps that are contained inother solution maps. For detailed membership tests we refer to [3].
OMPUTING ALL AFFINE SOLUTION SETS OF BINOMIAL SYSTEMS 3 Computational Experiments
The polynomial equations of adjacent minors are defined in [11, page 631]: x i,j x i +1 ,j +1 − x i +1 ,j x i,j +1 = 0 , i = 1 , , . . . , m − , j = 1 , , . . . , n − . For m = 2 , the solution set is puredimensional of degree n and of dimension n − ( n −
1) = n + 1 , the number of irreduciblecomponents of X equals the n th Fibonacci number [20, Theorem 5.9].For a pure dimensional set, we restrict the enumeration: for every variable we set tozero, one equation has to vanish as well. Table 1 shows the comparison with a witness setconstruction, computed with version 2.3.70 of PHCpack [21]. Note that our method returnsthe irreducible decomposition, which is more than just a witness set. This system is one ofthe benchmarks in [4], but neither Bertini [5] nor Singular [8] can get as far as our method. n n − Table 1.
The construction of a witness set for all adjacent minors of ageneral 2-by- n matrix requires the tracking of n − paths and is much moreexpensive than the combinatorial search. For n from 3 to 21 column 3 liststimes in seconds on one core at 3.49GHz for the combinatorial search andtimes ( < s e c o n d s o n . G H z I n t e l X a n d . G B m e m o r y times on the adjacent 2-by-2 minors of 2-by-n matrixmonomial mapswitness sets Table 2 shows timings of the binomialCellularDecomposition in the
Binomials [16]package of Macaulay2 [17] applied to the ideal defined by the adjacent minors. n Table 2.
CPU time in seconds on one 3.49GHz core on the adjacent minors.
DANKO ADROVIC AND JAN VERSCHELDE
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