| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula> be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>:</mo><mo>=</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo><mo>∪</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> for a non-empty open subset of <i>X</i>. For any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula>, the rank <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mrow><mi>X</mi><mo>,</mo><mi>w</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>q</i> with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the minimum of all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>a</mi><mo>∈</mo><mi>S</mi></mrow></msub><mi>w</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>S</i> is a finite subset of <i>X</i> spanning <i>q</i>. We have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mrow><mi>X</mi><mo>,</mo><mi>w</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo><</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> for all <i>q</i>. We discuss this definition and classify extremal cases of pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We give upper bounds for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mrow><mi>X</mi><mo>,</mo><mi>w</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> (twice the generic rank) not depending on <i>w</i>. This notion is the generalization of the case in which the cost-function <i>w</i> is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors.
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