@inproceedings{10.1145/1390768.1390790,
    Abstract = {Suppose the polynomials f and g in K[x1,...,xr] over the field K are determinants of non-singular m x m and n x n matrices, respectively, whose entries are in K ∪ x1,...,xr. Furthermore, suppose h = f/g is a polynomial in K[x1,..., xr]. We construct an s x s matrix C whose entries are in K ∪ x1,...,xr, such that h = det(C) and s = γ (m+n)6, where γ = O(1) if K is an infinite field or if for the finite field K = F{q} with q elements we have m = O(q), and where γ = (logq m)1+o(1) if q = o(m). Our construction utilizes the notion of skew circuits by Toda and WSK circuits by Malod and Portier. Our problem was motivated by resultant formulas derived from Chow forms.Additionally, we show that divisions can be removed from formulas that compute polynomials in the input variables over a sufficiently large field within polynomial formula size growth.},
    Address = {New York, NY, USA},
    Author = {Kaltofen, Erich and Koiran, Pascal},
    BookTitle = {Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation},
    File = {Expressing a Fraction of Two Determinants as a Determinant - skew.pdf},
    ISBN = {9781595939043},
    Keywords = {algebraic complexity theory, toda's skew circuits, formula complexity, valiant's universality of determinants, strassen's removal of divisions},
    Location = {Linz/Hagenberg, Austria},
    Pages = {141--146},
    Publisher = {Association for Computing Machinery},
    Series = {ISSAC '08},
    Title = {Expressing a Fraction of Two Determinants as a Determinant},
    URL = {https://doi.org/10.1145/1390768.1390790},
    Year = {2008},
    bdsk-url-1 = {https://doi.org/10.1145/1390768.1390790},
    date-added = {2022-10-22 08:13:11 +0200},
    date-modified = {2022-10-22 08:13:11 +0200},
    file-2 = {Expressing a Fraction of Two Determinants as a Determinant - 1390768.1390790.pdf},
    numpages = {6},
    doi = {10.1145/1390768.1390790}
}

@inproceedings{10.1145/1390768.1390790, Abstract = {Suppose the polynomials f and g in K[x1,...,xr] over the field K are determinants of non-singular m x m and n x n matrices, respectively, whose entries are in K ∪ x1,...,xr. Furthermore, suppose h = f/g is a polynomial in K[x1,..., xr]. We construct an s x s matrix C whose entries are in K ∪ x1,...,xr, such that h = det(C) and s = γ (m+n)6, where γ = O(1) if K is an infinite field or if for the finite field K = F{q} with q elements we have m = O(q), and where γ = (logq m)1+o(1) if q = o(m). Our construction utilizes the notion of skew circuits by Toda and WSK circuits by Malod and Portier. Our problem was motivated by resultant formulas derived from Chow forms.Additionally, we show that divisions can be removed from formulas that compute polynomials in the input variables over a sufficiently large field within polynomial formula size growth.}, Address = {New York, NY, USA}, Author = {Kaltofen, Erich and Koiran, Pascal}, BookTitle = {Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation}, File = {Expressing a Fraction of Two Determinants as a Determinant - skew.pdf}, ISBN = {9781595939043}, Keywords = {algebraic complexity theory, toda's skew circuits, formula complexity, valiant's universality of determinants, strassen's removal of divisions}, Location = {Linz/Hagenberg, Austria}, Pages = {141--146}, Publisher = {Association for Computing Machinery}, Series = {ISSAC '08}, Title = {Expressing a Fraction of Two Determinants as a Determinant}, URL = {https://doi.org/10.1145/1390768.1390790}, Year = {2008}, bdsk-url-1 = {https://doi.org/10.1145/1390768.1390790}, date-added = {2022-10-22 08:13:11 +0200}, date-modified = {2022-10-22 08:13:11 +0200}, file-2 = {Expressing a Fraction of Two Determinants as a Determinant - 1390768.1390790.pdf}, numpages = {6}, doi = {10.1145/1390768.1390790} }