On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

Herrero, María Isabel; Jeronimo, Gabriela; Sabia, Juan

doi:10.1007/s00454-018-0025-x

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“…A. Khovanskii informed the author that he had obtained (but did not publish) a bound equivalent to (43) which reduces the computation of generic intersection multiplicity to the convenient case. [HJS17] lists some other formulae for generic intersection mulitplicity in the general case.…”

Section: Notes and Referencesmentioning

confidence: 99%

How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

Mondal

2018

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Introduction 2. The bound 3. Derivation of the formuale for the bound 4. Other formulae for the bound 5. Examples motivating the non-degeneracy conditions 6. Non-degeneracy conditions 7. Proof of the non-degeneracy conditions 8. Open problems 9. Notes and references PrefaceIn this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k." The results coming out of this approach give the number of solutions for "generic" systems in terms of their Newton polytopes, and explicitly characterize what makes a system "generic." The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic systems on the "torus" (k \ {0}) n . In the context of our problem, however, the natural domain of solutions is not the torus, but the affine space k n . There were a number of works to extend Bernstein's theorem to the case of affine space, and recently it has been completely resolved. The aim of this book is to present these results in a coherent way. We start from the beginning, namely Bernstein's beautiful theorem which expresses the number of solutions of generic systems in terms of the mixed volume of their Newton polytopes. In fact one goal of this text is to give a complete proof of Benstein's theorem (over arbitrary algebraically closed fields) which is as elementary as possible. We also describe some results, both classical and new, on the theory of Milnor numbers of hypersurface singularities which can be obtained by similar techniques. Care was taken to make this book as elementary as possible. In particular, we develop all the necessary results from toric geometry and intersection theory. This book should be accessible to any student after a first course in algebraic geometry.The natural domain of solutions of systems of polynomials over a field k is not the torus (k * ) n , but the affine space k n . There are at least two different ways to extend Bernstein's theorem to k n . The approach motivated by the polynomial homotopy method for solving polynomial systems is as follows: given polynomials f 1 , . . . , f n , one starts with a deformed system f 1 = c 1 , . . . , f n = c n with nonzero c j . For generic 9 The Milnor number is an invariant of a singularity, see section 6.2. 10 A. Khovanskii gives a summary of Minding's approach in [BZ88, Section 27.3]; an English translation of [Min41] by D. Cox and J. M. Rojas appears in [GK03].11 The mixed volume is the canonical multilinear extension (as a functional on convex bodies) of the volume to n-tuples of convex bodies in R n , see section 3.2 for a precise description. 14 The bound from Bernstein's theorem and the sufficiency of ( * ) for the bound can be established without much difficulty (and also very elegantly!) using the general machinery of intersection theory (see e.g. [Ful93, Section...

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Section: Notes and Referencesmentioning

confidence: 99%

How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

Mondal

2018

Preprint

View full text Add to dashboard Cite

show abstract

On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

Cited by 1 publication

References 23 publications

How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

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