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12
Tropical arithmetic & algebra of tropical matrices
, 2005
"... The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discusse ..."
Abstract

Cited by 9 (6 self)
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The purpose of this paper is to study the tropical algebra – the algebra over the tropical semiring. We start by introducing a new approach to arithmetic over the maxplus semiring which generalizes the former concept in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward the characterization of the tropical inverse matrix which is eventually attained. Further development yields the notion of tropical normalization and the principle of basis change in the tropical sense.
Supertropical algebra
, 2007
"... Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more gener ..."
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Cited by 5 (3 self)
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Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more general situation over an arbitrary semiring, and develop a theory which contains the analogs of the basic theorems of classical commutative algebra (such as the Euclidean algorithm and the Hilbert Nullstellensatz), as well as some results without analogs in the classical theory, such as generation of prime ideals of the polynomial semiring by binomials. Examples are also given to show how this theory differs from classical commutative algebra.
Tropical Scaling of Polynomial Matrices
, 905
"... Abstract The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in p ..."
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Cited by 1 (0 self)
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Abstract The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.
unknown title
, 2007
"... Dynamic and static limitation in multiscale reaction networks, revisited ..."
unknown title
, 2008
"... Dynamic and static limitation in multiscale reaction networks, revisited ..."
unknown title
, 2007
"... Dynamic and static limitation in multiscale reaction networks, revisited ..."
unknown title
, 2007
"... Dynamic and static limitation in multiscale reaction networks, revisited ..."
Contents
, 806
"... Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more gener ..."
Abstract
 Add to MetaCart
Abstract. We develop the algebraic polynomial theory for “supertropical algebra, ” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of ghost elements, which play the key role in our structure theory. Here, we work in a slightly more general situation over an arbitrary semiring, and develop a theory which contains the analogs of the basic theorems of classical commutative algebra (such as the Euclidean algorithm and the Hilbert Nullstellensatz), as well as some results without analogs in the classical theory, such as generation of prime ideals of the polynomial semiring by binomials. Examples are also given to show how this theory differs from classical commutative algebra.