{"history":[],"article":{"id":"76a54997-b006-415d-955b-8109fb3816c1","title":"Regular chain","slug":"regular-chain","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), and more specifically in [computer algebra](https://bliptext.com/articles/computer-algebra) and [elimination theory](https://bliptext.com/articles/elimination-theory), a **regular chain** is a particular kind of triangular set of [multivariate polynomials](https://bliptext.com/articles/multivariate-polynomial) over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one.\nThe condition that a triangular set must satisfy to be a regular chain is that, for every k, every common zero (in an [algebraically closed field](https://bliptext.com/articles/algebraically-closed-field)) of the k first polynomials may be prolongated to a common zero of the\n(k + 1)\nth polynomial.\nIn other words, regular chains allow solving [systems of polynomial equations](https://bliptext.com/articles/systems-of-polynomial-equations) by solving successive univariate equations without considering different cases.\nRegular chains enhance the notion of [Wu's characteristic sets](https://bliptext.com/articles/wu-s-method-of-characteristic-set) in the sense that they provide a better result with a similar method of computation.\n\n# Introduction\nGiven a [linear system](https://bliptext.com/articles/system-of-linear-equations), one can convert it to a [triangular system](https://bliptext.com/articles/triangular-matrix) via [Gaussian elimination](https://bliptext.com/articles/gaussian-elimination).\nFor the non-linear case, given a [polynomial system](https://bliptext.com/articles/system-of-polynomial-equations) F over a field, one can convert (decompose or triangularize) it to a finite set of triangular sets, in the sense that the [algebraic variety](https://bliptext.com/articles/algebraic-variety) V(F) is described by these triangular sets.\nA triangular set may merely describe the empty set.\nTo fix this degenerated case, the notion of regular chain was introduced, independently by Kalkbrener (1993), Yang and Zhang (1994).\nRegular chains also appear in Chou and Gao (1992).\nRegular chains are special triangular sets which are used in different algorithms for computing unmixed-dimensional decompositions of algebraic varieties.\nWithout using factorization, these decompositions have better properties that the ones produced by [Wu's algorithm](https://bliptext.com/articles/wu-s-method).\nKalkbrener's original definition was based on the following observation: every irreducible variety is uniquely determined by one of its [generic points](https://bliptext.com/articles/generic-point) and varieties can be represented by describing the [generic points](https://bliptext.com/articles/generic-point) of their irreducible components.\nThese generic points are given by regular chains.\n\n# Examples\nDenote **Q** the rational number field.\nIn **Q**[x1, x2, x3] with variable ordering x1 < x2 < x3,\nis a triangular set and also a regular chain.\nTwo generic points given by T are (a, a, a) and (a, −a, a) where a is transcendental over **Q**.\nThus there are two irreducible components, given by { x2 − x1, x3 − x1 } and { x2 + x1, x3 − x1 }, respectively.\nNote that: (1) the [content](https://bliptext.com/articles/content-algebra) of the second polynomial is x2, which does not contribute to the generic points represented and thus can be removed; (2) the [dimension](https://bliptext.com/articles/dimension) of each component is 1, the number of free variables in the regular chain.\n\n# Formal definitions\nThe variables in the polynomial ring\nare always sorted as x1 < ⋯ < xn.\nA non-constant polynomial f in R can be seen as a univariate polynomial in its greatest variable.\nThe greatest variable in f is called its main variable, denoted by mvar(f).\nLet u be the main variable of f and write it as where e is the degree of f with respect to u and a_e is the leading coefficient of f with respect to u.\nThen the initial of f is a_e and e is its main degree.\nA non-empty subset T of R is a triangular set, if the polynomials in T are non-constant and have distinct main variables.\nHence, a triangular set is finite, and has cardinality at most n.\nLet T = {t1, ..., ts} be a triangular set such that mvar(t1) < ⋯ < mvar(ts),\nh_i be the initial of ti and h be the product of hi's. Then T is a regular chain if where each [resultant](https://bliptext.com/articles/resultant) is computed with respect to the main variable of ti, respectively.\nThis definition is from Yang and Zhang, which is of much algorithmic flavor.\nThe quasi-component W(T) described by the regular chain T is the set difference of the varieties V(T) and V(h).\nThe attached algebraic object of a regular chain is its saturated ideal\nA classic result is that the [Zariski closure](https://bliptext.com/articles/zariski-closure) of W(T) equals the variety defined by sat(T), that is,\nand its dimension is n − |T|, the difference of the number of variables and the number of polynomials in T.\nIn general, there are two ways to decompose a polynomial system F.\nThe first one is to decompose lazily, that is, only to represent its [generic points](https://bliptext.com/articles/generic-point) in the (Kalkbrener) sense,\nThe second is to describe all zeroes in the [Lazard](https://bliptext.com/articles/daniel-lazard) sense,\nThere are various algorithms available for triangular decompositions in either sense.\n\n# Properties\nLet T be a regular chain in the polynomial ring R.\n\n# Further references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:02:25.588Z","updated_at":"2025-01-26T11:02:25.588Z","search_vector":"'chain':2 'regular':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), and more specifically in [computer algebra](https://bliptext.com/articles/computer-algebra) and [elimination theory](https://bliptext.com/articles/elimination-theory), a **regular chain** is a particular kind of triangular set of [multivariate polynomials](https://bliptext.com/articles/multivariate-polynomial) over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one.\nThe condition that a triangular set must satisfy to be a regular chain is that, for every k, every common zero (in an [algebraically closed field](https://bliptext.com/articles/algebraically-closed-field)) of the k first polynomials may be prolongated to a common zero of the\n(k + 1)\nth polynomial.\nIn other words, regular chains allow solving [systems of polynomial equations](https://bliptext.com/articles/systems-of-polynomial-equations) by solving successive univariate equations without considering different cases.\nRegular chains enhance the notion of [Wu's characteristic sets](https://bliptext.com/articles/wu-s-method-of-characteristic-set) in the sense that they provide a better result with a similar method of computation.\n\n# Introduction\nGiven a [linear system](https://bliptext.com/articles/system-of-linear-equations), one can convert it to a [triangular system](https://bliptext.com/articles/triangular-matrix) via [Gaussian elimination](https://bliptext.com/articles/gaussian-elimination).\nFor the non-linear case, given a [polynomial system](https://bliptext.com/articles/system-of-polynomial-equations) F over a field, one can convert (decompose or triangularize) it to a finite set of triangular sets, in the sense that the [algebraic variety](https://bliptext.com/articles/algebraic-variety) V(F) is described by these triangular sets.\nA triangular set may merely describe the empty set.\nTo fix this degenerated case, the notion of regular chain was introduced, independently by Kalkbrener (1993), Yang and Zhang (1994).\nRegular chains also appear in Chou and Gao (1992).\nRegular chains are special triangular sets which are used in different algorithms for computing unmixed-dimensional decompositions of algebraic varieties.\nWithout using factorization, these decompositions have better properties that the ones produced by [Wu's algorithm](https://bliptext.com/articles/wu-s-method).\nKalkbrener's original definition was based on the following observation: every irreducible variety is uniquely determined by one of its [generic points](https://bliptext.com/articles/generic-point) and varieties can be represented by describing the [generic points](https://bliptext.com/articles/generic-point) of their irreducible components.\nThese generic points are given by regular chains.\n\n# Examples\nDenote **Q** the rational number field.\nIn **Q**[x1, x2, x3] with variable ordering x1 < x2 < x3,\nis a triangular set and also a regular chain.\nTwo generic points given by T are (a, a, a) and (a, −a, a) where a is transcendental over **Q**.\nThus there are two irreducible components, given by { x2 − x1, x3 − x1 } and { x2 + x1, x3 − x1 }, respectively.\nNote that: (1) the [content](https://bliptext.com/articles/content-algebra) of the second polynomial is x2, which does not contribute to the generic points represented and thus can be removed; (2) the [dimension](https://bliptext.com/articles/dimension) of each component is 1, the number of free variables in the regular chain.\n\n# Formal definitions\nThe variables in the polynomial ring\nare always sorted as x1 < ⋯ < xn.\nA non-constant polynomial f in R can be seen as a univariate polynomial in its greatest variable.\nThe greatest variable in f is called its main variable, denoted by mvar(f).\nLet u be the main variable of f and write it as where e is the degree of f with respect to u and a_e is the leading coefficient of f with respect to u.\nThen the initial of f is a_e and e is its main degree.\nA non-empty subset T of R is a triangular set, if the polynomials in T are non-constant and have distinct main variables.\nHence, a triangular set is finite, and has cardinality at most n.\nLet T = {t1, ..., ts} be a triangular set such that mvar(t1) < ⋯ < mvar(ts),\nh_i be the initial of ti and h be the product of hi's. Then T is a regular chain if where each [resultant](https://bliptext.com/articles/resultant) is computed with respect to the main variable of ti, respectively.\nThis definition is from Yang and Zhang, which is of much algorithmic flavor.\nThe quasi-component W(T) described by the regular chain T is the set difference of the varieties V(T) and V(h).\nThe attached algebraic object of a regular chain is its saturated ideal\nA classic result is that the [Zariski closure](https://bliptext.com/articles/zariski-closure) of W(T) equals the variety defined by sat(T), that is,\nand its dimension is n − |T|, the difference of the number of variables and the number of polynomials in T.\nIn general, there are two ways to decompose a polynomial system F.\nThe first one is to decompose lazily, that is, only to represent its [generic points](https://bliptext.com/articles/generic-point) in the (Kalkbrener) sense,\nThe second is to describe all zeroes in the [Lazard](https://bliptext.com/articles/daniel-lazard) sense,\nThere are various algorithms available for triangular decompositions in either sense.\n\n# Properties\nLet T be a regular chain in the polynomial ring R.\n\n# Further references","revisionCount":0},"hasMore":false,"nextCursor":null}

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